Differential Equations from the Group Standpoint

L. E. Dickson

The Annals of Mathematics, 2nd Ser., Vol. 25, No. 4. (Jun., 1924), pp. 287-378.

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 DIFFERENTIAL EQUATIONS FROM THE GROUP

STANDPOINT.

Introduction.

The various classic devices for the integration of differential equations

may be explained simply from a single standpoint-that of infinitesimal
transformations leaving the equations invariant. What is still more im-
portant than this unification of diverse known methods, infinitesimal trans-
formations furnish us a new tool, likely to succeed when the ordinary methods
fail, since they enable us to take into account vital information ignored
by the ordinary methods. In fact, the new method does not confine attention
to the differential equation and ignore the data of the problem of which
the equation is an analytic formulation, but makes use of the data itself
in order to obtain one or more infinitesimal transformations leaving the
equation invariant. Accordingly the new method is most readily applied
successfully to differential equations arising in geometry or mechanics. Why
bother with a dead equation whose origin is unknown or has been concealed?
Although no previous acquaintance with differential equations is pre-
supposed, the paper is not proposed as a substitute for the usual intro-
ductory course, but rather to provide a satisfactory review ab initzo and a t
the same time to present the unifying and effective method based on groups.
The important topic of difFerentia1 invariants is given considerable attention
a t appropriate places throughout the paper. Application is made in $ 54
to the congruence of plane curves and their intrinsic equations.
Finally, we obtain in 5 55 a complete set of functionally independent
covariants and invariants of the general binary form and deduce the
fact,s that every polynomial invariant of the binary quadratic or cubic
form is a polynomial function of its discriminant, while every polynomial
invariant of a quartic form is a polynomial function of two specified in-
variants. This method of attack provides an easy introduction to the com-
plicated algebraic theory of invariants as well as the relation between that
subject and the topic of functionally independent invariants.
The writer is greatly indebted to the founder of the theory of continuous
groups, Sophus Lie, whose lectures he attended in 1896. Numerous valuable
suggestions on the manuscript were received from Professor Bliss. It has
been used in classes by Dr. Barnett and the writer.
 L . E. DICKSON .

Table of Contents.

Section Page

1-3 Translations. rotations. transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .289 .

4 Product of two translations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290

5 Inverse of a translation. identity trsnsformatiou . . . . . . . . . . . . . . . . . . . . . . . . . . . 291

6 Groups of transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 . 92

7 Groups found by integrating differential equations . . . . . . . . . . . . . . . . . . . . . . . . . 293

8 Infinitesimal transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295

9 Every one-parameter group is generated by an infinitesimal transformatiori . . . 297

10 Equivalence of two one-parameter groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299

11-12 Invariant points. Path curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .300

13 Conespondillg ordinary and partial differential equations . . . . . . . . . . . . . . . . . . . 302

14 First criterion for the invariance of a differential equation of the first order

undei a group .........................................................303

15-16 Finding an integrating factor, problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .304

17 Infinitude of groups leaving invariant a differential equation of the &st order 308

18 Geometrical interpretation of Lie's integrating factor . . . . . . . . . . . . . . . . . . . . . . .308

19-20 Parallel curves, isothermal curves ........................................310

21 Commutator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .313

22 A second criterion for the invariance of a differential equation under an in-

finitesimal transformation ................................................314

23 Group of extended transfonnationu .......................................315

24-26 Invariants. differential invariants .........................................316

27 Invariant equations .....................................................319

28 A third criterion for the invariance of a differential equation of the first order

under Uf ............................................................ 320

29 Introduction of new variables in a linear partial differential expression ...... 3%

80 Determination of all dserential equations of the 6rst order invariant under a

given infinitesimal transformation, table .................................. 322

31-34 Complete system of linear partial differential equations ..................... 3.25

35 Standard methods of solving a complete systeni of two partial differential

equations in three variables ............................................. 331

36-36 Solution of one partial differential equation iuvariant under m infinitesimal

transformation ..........................................................333

39 Jacobi's identity ........................................................336

40-42 Solation of one partial Mereutial equation invariant under two infinitesimal

transformations ......................................................... 336

43 Second extension of an infinitesimal transformation ........................ 344

44 Differential invariants of the second order ................................ 345

45-46 Integration of differential equations of the second order invariant under one

infinitesimal transformation. table ........................................ 346

47-48 Number of linearly independent infinitesimal transformations leaving y "=w(z, y,y')

invariant .............................................................. 352

49 Integration of y" = w invariant under two infinitesimal transformations ...... 354

50 First extension of a commutator .......................................... 357

51-52 Closed system of infinitesimal transformations leaving y" = w invariant ...... 358

53 Integration of y" = w .................................................. 364

54 Differential invariauts and the congruence of plane curves .................. 367

55 Algebraic invariants and cuvarianta ....................................... 369

 DIFFERENTIAL EQUATIONS FROM THE GROUP STANDPOINT. 289

CHAFTERI.

One-parameter groups of transformations.

The object of this chapter is to define and illustrate the concepts trans-
formation and group of transformations whose equations involve a single

/'
parameter. Each such group is generated by an infinitesimal transformation.
The latter will prove to be more convenient for the subsequent applications
than the group.
1. Translations. In analytic h Y
geometry we interpreted the pair I

of equations p -"-'--"'
4

Pmx
(1) Xl =x+4, Yl = y + 3
as relations between the coordinates
x, y of any point Preferred to one 13

pair of rectangular axes Ox, Oy,

Xl

01 4
and the coordinates x,, y, of the
same point P referred to another Fig. 1.

pair of rectangular axes O1x,, 0, y,, parallel to the former axes respectively,

and such that 0 has the codrdinates (4,3) with respect to the axes O1x,, (I,yl.

On a sheet of transparent paper covering Fig. 1, draw traces of the lines

O1xl and 0, y, and of the point P having the coordinates x,, yl referred

to those axes. Move the transparent sheet without rotation (so that O,xl

remains parallel to itself during the motion) until the trace of 0, covers 0

and hence the trace of Olxl covers Ox. Under this translation of the

transparent sheet, the trace of P moves to the point P, having the co-

ordinates x,, y, referred to the axes Ox, Oy.

This leads us to the following new interpretation of equations (1). We

ignore henceforth the axes O,x, and 0, y, and regard x, y as the coordinates

of one point P and x,, yl as the coardinates of a new point PI,each referred

to the same axes Ox, Oy. In view of equations (I), the line PP,is parallel

to 0,O snd of the same length 5 as 0,O. Under our new interpretation,

fo~mulas(1) accomplisLl a translation of all points of the plane through

a distance 5 in a direction parallel to 0,O.

2. Rotations. If x, y and x,, y, are the coBrdinates of the same

point P referred to two pairs of rectangular axes Ox, Oy and Ox,, Oy,,
respectively, such that t is the positive angle measured counter-clockwise
from Ox, to Oz, it is proved in analytic geometry that

(2) x1 = xcost-ysint, y, = xsifit+yccst.

290 L. E. DICKSON.

On a sheet of transparent paper covering Fig. 2, draw traces of the

lines Ox,. Oy, and of the point P having the coordinates x,, y, referred
to those axes. Rotate the transparent
Y sheet counter-clockwise about 0 through
angle t, so that the trace of 0% will
now cover Ox. Then the trace of
P : (x, y) is rotated to the position P,
shown in Fig. 2.
This leads us to the following new
P ~ T . ~ i) nterpretation. When both points (x, y)
and (x,, y,) are referred to the same
0
rectangular axes Ox, Oy, equations (2)
define a rotation of all points of the
plane about the origin 0 counter-
clockwise through angle t.
3. Transformations. Generalizing
from the translation (1) and the rotation ( 2 ) , we shall say that any pair
of equations of the type

defines a tramfmmation of all points of the plane, provided g and h are

independent functions of z,y. It is again understood that the arbitrary
initial point (x, y) and its transformed point (q,yl) are both referred to
the same pair of rectangular axes.
4. Product of two translations. If a is a positive constant, the pair
of equations
T,: zl=x+a, yl=y

represents a translation through a distance a parallel to the x-axis and

toward its positive direction. Let

represent another such translation; it carries the point (r, y) to the

+
point (x b. y). Since x and y are arbitrary an2 hence may be assigned
the values z, and y,, we see that T b carries the point (x,, y,) to the
+
point (x, b, yl), which will be designated also by (Q, y,). We may there-
$ore express T b in the form
 DIFFERENTIAL EQUATIONS FROM THE GROUP STANDPOINT. 29 1
Since T, carries the point (x, y) to the point (xl, yl), which Tb carries
to the point (x,, y,), the combined effect of the two translations acting in
succession is to carry the point (a,y) to the point (x,, yr) such that

This replacement of the point (x, y) by the point (x,, y,) or (xf a+b, y)
is also accomplished by the single translation

This latter translation is called the p?.odt~tof T, and T b , taken in this

order, and is designated by ToTb.The word "product" is here a technical
term having the sense of "compound" or "resultant". Our conclusion,

is also evident from mechanical considerations. The effect of two successive

translations through the distances a and b, each in a direction parallel to
the positive x-axis, is the same as that produced by the translation through
the distance a + b parallel to the positive x-axis.
5. Inverse of a translation, i d e n t i t y transformation. In the trans-
lation Tbwe now permit b to take the negative value -a. Then the pair
of equations
T-@: xl = x-a, YI = Y

represents a translation through a distance a parallel to the x-axis, but

toward its negative direction. The effect of T-a is therefore just the
reverse of that of Ta; it undoes what was done by Ta,or effects the
"return trip". For, Ta translates the point (x, y) to (x+ a, y), while
+
T-, translates the latter point to (x a-a, y), which is the initial
point (x, y). For these reasons we shall call T-a the inverse of To and
designate it by 27'.
The product ~ a , ~ a - ' is xl = x, y1 = y, which is called the identity
transformation and designated by I, since the transformed point (xl, yl)
under I is always identical with the initial point. Also, T
, T, = I. Thus

Similar ideas are involved in the definition of the inverse trigonometric functions. For
example, if the sine of angle A has the value v , then A is called the inverse sine uf v,
and we write d = sin-' a. Thus sin(sin-* 1 ) ) = v.
292 L. E. DICKSON.

6 . Groups of transformations. The set of translations

Ta: X I = X ~ $ ~ > Y1 = Y,

obtained hen a ranges over all integral values (i. e., the positive and
negative whole numbers and zero), is said to constitute a g1'021p since it
has the follo~\~ing two properties.
(i) The product of any two (distinct or coincident) transformations of
the set is itself ore of the transformations of the set.
(ii) The inverse of anj- transformation of the set is itself a transformation
of the set.
For, if a and Z, are any two integers (distinct or equal),

and c and - a are also integers. This group is called discontitzztous

since the parameter n has the discontinnous range of all integral values.
Discontinuous groups are useful in the theory of numbers, the theory of
elliptic modular fnnctions, and Galois' theory of equations. They will be
excluded ,in what follows.
But the set of translations Ta in which a ranges over all real numbers
forms n ronti~zuoti.~one-l~aramstwgroup since the parameter a has a con-
tinuous range of values and since the formulas for T, may be converted into
the formulas for T, by cor,tinuous variation of the parameter from nl to a,.
The second condition of this definition of a continuous group is not
satisfied for the mixed g~oztp composed of all the Ta together with the
transformations
R, : xl=x+a, yl=-y,

where again c( ranges over all real numbers. We have a group since

R e shall not employ mixed groups in this paper.

The set of rotations (2) in which t ranges over all real numbers is
a continuous one-parameter group. For, the product of rotation (2) by
the rotation
r2 = X I cos t' -y1 sin t', +
y, = x, sin t' yl cos t'

is readily verified to be the rotation about the origin counter-clockwise

+
through angle t t', while the inverse of (2) is the rotation through the
angle - t.
 DIFFERENTIAL EQUATIONS FROM TEE GROUP STANDPOINT. 293

Again, all similarity transformations

form a continuous one-parameter group. The same is true of

The next discussion shows how to find as many continuous one-parameter

groups as we please, in fact, one group fron~ each pair of functions
Z(x, y), q(x, y), which are continuous in a certain region of the sy-plane.
7. Groups found by integrating differential equations. We employ
the system of differential equations

These imply the equation in two variables

which is known* to possess one and only one integral curve v (z,, y,)
passing through any given point (z, y ) and defined for all points (x,, yl)
c -
of a certain region containing (x, y). Evidently c = v(s. y).
In order to integrate the system (3), we solvef v(rl, yl) = c for yl in
terms of 2, and c, substitute in the first equation (3), and integrate. Let
the result be 20(x1, c)- t = c', where c' is an arbitrary constant. Replacing
c by its value v(x,, y,), we obtain a result of the form i((z,,yl)-t = c'.
Without loss of generality we may assume that t = 0 when q = x, y, = y.
In view of these initial values, we have

Let the solved form of these equations (5) be

* Picard, Trait6 d'.inalyse, 11, 1893, pp. 292, 301, 304; 111, 1896, p. 88. Bliss, Princeton
Colloquium Lectures, 1913, p. 86.
PI11 case .v is independent of y,, we interchange the rbles of X I , y, in what follows.
294 L. E. DICKSON.

in which the functions are defined for values of t which are sufiiciently
small numerically.
We shall prove that these transformations (6) form a group by verifying
that they have the two properties listed in 5 6. The product of (6) by
a second transformation
(7) Ice = O(x1, y,, t'), Y* = P(x1, Yl, tl)
of the same set is the follo.wing transformation:

To simplify the proof, we return from (6) to the equivalent equations (EJ),
and likewise from (7) to the equivalent equations

whose solved form is (7) for the same reason that (6) is the solved form
of (5). Eliminating a,y, between (5) and (9), we evidently get

whose solved form is (8). Hence the product of two transformations (6)
whose parameters t and t' are sufficiently small numerically is a trans-
formation (6) with the parameter t t'. +
+
Since (10) is the identity transformation if t t' = 0, the inverse of (6)
is derived from (6) by replacing t by - t.
If me take X = zt(x, y) and I.'= v(x, y) as new variables, we obtain
from (5) the translation
(11) X1=X+t, Y1=Y.
THEOREM. When the parametw t is restricted to values sufMently small
numerically, the solzbtions (6) of the system of differential equations (3) form
a continuous orhe-parameter group, which can be reduced to the group of
trarrslations (11) by the introduction of new variables.
For example, if € = - y , , 7 = x,, (4) becomes x,dxl + yldyl = 0,which has the
integral v = v x : + y; = c. The second equation (3) may be written in the form

~ Y I = dt, sin-I =t + c', y l = c sin (t + c').

l / q C
Then
I= = c COB (t + c'), u E tan-'-
Yl
*1
= t+c',
 DIFFEREWTIAL EQUATIONS FROM THE GROUP STANDPOINT. 295
Employing polar coardinates 0, p, we get

which are equations of type (11). This group of rotations may be given the form (2) by
solviug equations (5') for zl,
yl.
8. Infinitesimal transformations. In view of the equations (6) of
a group, any function f(xl, y,) of x, and y, is a function of x, 2, and t.
We have

Naking use of (3), we get

When t = 0, xl becomes x, and yl becomes y. Hence

We deduced (12) frcm the pair of equations (3). Conversely, (12) im-
plies (3), as seen by taking f(x,,y,) to be x, and y, in turn. Hence (1 2)
is a convenient synthesis of tne pair of equations (3). By the integration
of (3) we obtained the equations (6) of a group. Accordingly we shall
say that equations (3), or their synthesis (12), generate the equations (6)
of the group. Dropping the subscripts 1 in (12), we obtain the expressioa
denoted by Uf in (13). With Sophus Lie. we shall speak of Uf as the
(symbol of the) in$nitesimnl transformation which generates the eqz~ations(6)
of tlze transformations of the group. In brief, we shall say that Uf gene-
rates the group.
Xot only does the symbol Uf give the pair of functions S(x. y), ~ ( zy),
and hence the differential equations (3) whose integration yields the equa-
tions (6) of the group, but conversely the equations (6) determine Uf since

of which (13) is the synthesis.

 -.
296 L. E. DICKSON.

For 6 - y, 7 x, the infinitesimal transformation is

By the example in 5 7, uf generates the group of all rotations (2) about the origin.
Starting with that group, we readily find the preceding infinitesimal transformation by
differentiation. For, by (2) and (14),

- -- - x
ax,
dt sin t - y c o s t = -y,,

By Maclaurin's theorem any function fl -

-dd=Yl
tX I ; ;
.-
= -9,

f (x,, y,), which is regular

a t the point (x,, y,), may be expanded into a series in t for t sufficiently
small numerically :
7 EE x.

Denote the second member of (12) by U1, which therefore becomes Uf

(x, y ) in (13) when t = 0. Hence df, I d t = U, fl. Since is itself
a function of q,y,, the same formula gives

etc. Hence

where f denotes f (z, y). In particular, if we take f(x,, y,) to be x, and y,

in turn, we get

which give the equations, in the fmm of series, of the transformations of

the group genwated by Uf. Here t is restricted to the values for which
the series converge. We recall also the restriction on t in the equations (6)
of the group obtained by integrating differential equations. But these
restrictions are unimportant for the applications which depend upon the
transformations defined by the values of the parameter in the neighbor-
hood of t = 0.
By (13), U x = 6, U y = 7. I t is a common practice to employ the infinitesimal
notation d t for a value of t in the immediate neighborhood of t = 0, to neglect higher
powers of dt, to write (16) in the form
 DIFFERENTIAL EQUATIONS FROM THE GROUP STANDPOINT. 297
and to regard these as the equations of the infinitesimal transformation Uf. n i s serves
to expiain the historical origin of the latter term. However, we shall avoid the use of
infinitesimal quantities.
Employing the terminology explained here, we may state the theorem
of § 7 in the form in which it will be quoted later.
THEOREM. Every injinitesimal transformnfion on x and y generates (in
the sense of infegration) a c o n t i n u r n one-parameter group. After suitably
chosen new variables have been introdzcced, the group bb~comes the group of
translations x, = x + t, y, = y, generated By the injinitesinzal transfor-
mation af 1ax.
9. Every one-parameter group is generated b y an infinitesimal
transformation. We shall first treat the typical case of the group of
similarity transfornations

Sa : x1 = a x , yl = ay.

Since SaSa = Saa, the parameter of the product is the product of the
parameters of the component transformations, whereas it was their sum in
the case of the group of the transformations (6). Hence we cannot put
the transformations Sa into one-to-one correspondence with the trans-
formations of any group of type (6) by taking a = t. But this can be
done by taking a = et, where e is the base of natural logarithms, and t
takes only real values if we restrict a. to positive values, Then

and, by (14),1 = x, tj = y. The resulting infinitesimal transformation

generates the group (17), whose equations give the solutions of

subject to the initial conditions xl = x , yl = y, when t = 0.

Using similar ideas, we next treat any one-parameter group of trans-
formations
(18) X I = g(x,Y, a), yt = h(x, y, a).
298 L. E. DICKSON.

The product of (18) by any other transformation

of the group must be a third transformation

of the group. Hence there exists a function c = y (a, b) such that

identically in x, y, a, b: with a similar identity for h. We differentiate*

with respect to b, insert the abbreviations (18), and get

The group contains the identity transformation 1 given by the value u,

of the parameter a. For b = 6,(19) becomes I, and the product of (18)
by (19) becomes (18f7 whence c = a. Hence the last identity becomes

The final factor is unity when a = ao, since (18) is I f o r a = a,, whence
y ( a o , b) = b. Since the final factor is not zero identically, and contains
the parameter a continuously, we may denote it by l/w(a.). Denote the
left member of (22) by S(xl,yl), and the similar expression in h by q ( x l ,yl). Then

Introduce the new parameter

* W e assume that the two functions (19) may be differentiated with respect to b.
 DIFFERENTIAL EQUATIONS FROM THE GROUP STANDPOINT. 299

in place of a. Then
d t = w(a)da,
dxl = -
-
dt
dxl d a
cla
.-d t '

Lf a = h, then xl = x, y1 = y, t = 0, which are the initial values

employed in 5 7.
THEOREM.T he transformations (18) of any one-parameter group whose
identity transformation has the parameter a, are put into one-to-one corre-
spondence, by the introdt~ctirmof the new parameter (24), with the trans-
formations of the groztp generated by the in.nitesima2 transformation

These values of 5, 71 follow from (22) for a = a, since w(a,) = 1.

10. Equivalence of two one-parameter groups. Two groups

are called equivalent if and only if there exists a one-to-one correspondence

A = @(aj such that

identically in x, y, a. By (23), these imply

in which W, X , Y are the functions for the second group defined in the
same manner that w, 5, 9 were defined for the first group. Taking a = h,
where a, gives the identity transformation of the first group, we get
300 L. E. DICKSON.

where k is a constant. Conversely, if the last two identities hold, the

groups are equivalent under the correspondence k t = T, since from (3)
we get

dx1-_ -d-xl
- -d ~ l d Yl
dT kdt - ( 7 1 d=T - - k d t - Y(x1, ?/I).

THEOREM.TWOone-parameter groups are equivalent if and only if the

symbols of their illfinitesimal transformations di$er only by a constant factor.
11. Invariant points. A point may be invariant (left unchanged) by
all the transformations of a group. For example, the origin is invariant
under all of the transformations

If a point (x, y) remains unaltered by every defmed transfo~mation(6)

of a group, so that xl = x, y1 = y, for every permissible value of 2, then

evidently vanish a t (x, y). Hence by (3), f (2, y) and ~ ( xy)

, both vanish
a t (2, y). This also follows fro111 (16), which implies the converse.
THEOREM.A point (x, y) is invariant under the gvoup generated by the
i?z$nitesimal transformatirm

if and only if $ and q both vanish a t (z, y).

12. Path curves. By way of introduction, consider the group Q of
similarity transformations (27). Any point (G, yo), not the origin, is carried
by the various transformations of O into the various points (axo, aye)
whose locus is the straight line q y = yox. Noreover, any point (a1xO7atyo)
of the latter is carried by Q into points of the same line. For these
reasons, any straight line through the origin is called a path curve of
the group Q.
Consider any one-parameter group Q of transformations
 DIFFERENTIAL EQUATIONS FROM THE GROUP STANDPOINT. 301

whose identity transformation is given by the value a. of a. Through any

point (xo,yo), which is not invariant under Q, there passes one path curve Co
whose parametric equations are
(30) o , 4,
x = ~ ( x yo, Y = h ( X O , Yo, a),
and whose ordinary equation is found by eliminating a. For, if both of these
functions were independent of a (so that the elimination of a would be
impossible), they would be identical with their values xo, yo for a = ao,
whereas the point (G,yo) is not invariant under Q. That the curve Co passes
through (rc,, yo) is shown by taking a = a,.
If we apply to any point ( x , y) on Co any transformation Tt, of the group G,
we obtain a point ( x l ,yl) on Co. For, we may interpret (30) as a trans-
formation (29) of Q which carries (%, yo) to ( x , y). Since the product T a Ta
is a transformation T,of Q and carries (G,yo) to (x,, y,), we have

21 = g(x0, Yo, c), Y1 = (xo, Yo, 4,

whence the point ( x l , yl) is on the curve Co defined by (30).
If (x, y) and ( x ' , y') are any two points on Co, so that (xo,yo) is carried
to them by certain transformations Toand Ta' of O, then (x,y) is carried
to ( s f ,TJ')by the transformation Ta-' Tar of Q.
If the curve Co has a point (x', y') in common with another path curve of Q,

the last result shows that (x', y') is carried to any chosen point P on C,
by some transformation T of Q. But the point (xo,yo) is carried to (x', y')
by some transformation T of Q. Hence T ' T carries (xo,yo) to P, so that P
lies on Co since T'T is a transformation of Q. This proves that C,
coincides with Co.
THEOREM 1. Every one-parameter group Q has a family of path curves the
points on each of which are merely permuted by the transformations of G and
such that there is m e and only one path curve through each point of the plane
not a n invariant point.
Since the equation of any path curve is obtained by eliminating a between
the equatiolis (30), which may be interpreted as the equations (29) of the
general transformation of the group, and since the functions (29j are the
solutions of the system of differential equations (23), it follows (after dropping
the subscripts 1) that the path curves are the integral curves of
302 L. E. DICKSON.

and hence are v ( x , y) = c in the notation of 5 7. By differentiation,

Since d x and d y are proportional to F and q by (31),

Hence Uv r 0, where U f is the infinitesimal transformation (28) which

generates the group.
Conversely, if v (x,y) is a solution of U f = 0 , (12) shows that
d ( x y ) 1d t 0. Thus v (xl, yl) is independent of t and hence is equal
to its value v(z, y) for t = 0. Hence every trransformation of the group
replaces each point (x, y) of the curve v(x, y) = c by a point ( x l ,y,) on
the same curve, which is therefore a path curve.
THEOREM2. The path clkrves of the group generated by an infinitesimal
transformation U f are obtained by equating lo an arbitrary constant a
solution of Uf = 0 . The slope of a path curve at (x, y) is
EXAMPLES.For the infinitesimal rotation

a solution of Uf = 0 is x2+ y2. The path curves of the group of rotations about the
+
origin (5 8) are therefore circles m2 y2 = c. The slope at (x, y) is -x/y = 116
Consider the infinitesimal trausfomation

which generates the group of similarity transformations (9 9). Since a solution of Uf = 0

ie y/x, the path curves are the straight lines through the origin.
13. C o r r e s p o n d i n g ordinary and p a r t i a l differential equations.
If v(x, y) = c is an integral of the ordinary differential equation (31),
we saw that v is a solution of the corresponding linear partial differential
equation

Conversely, if a function v ( x , y), which is not a constant, is a solution

of (34), so that (33) holds, then the latter together with the result (32)
of differentiating v = c show that dx and dy are proportional to F and 7,
whence v = c is an integral of (31).
 DIFFERENTIALEQUATIONS FROM THE GROUP STANDPOINT. 303
THEOREM. If v(x, y ) = c is a n i?ategral of tlze ordinary diffe-rential
eqzsntion (31), then v is a solution of the rorresl~ondingZinearpartial differ-
ential eqzcntion (34), and conuersely.

CHAPTER11.
Ordinary differential equations of the first order.
The problem of integrating a differential equation will be reduced to the
problem of finding a one-parameter group which leaves the equation un-
altered. Such a group presents itself immediately when the eqnation is
linear or homogeneous or of certain other standard types, but especially
for differential equations arising in problems in geometry and mechanics
($8 16, 18--20). In the latter cases we do not attempt to integrate the
equation ignoring the data of the problem in which it arose, but rely on
the data to suggest a group which leaves the equation unaltered.
14. First criterion for the invariance of a differential equation
under a group. If the integral curves O ( x , y) = c of a differential
eqnation of the first order are merely permuted among themselves (or are
individually unaltered) by every transformation of a one-parameter group,
the differential equation is said to be invariant under the group. The
phrase "invariant under the group generated by the infinitesimal trans-
formation U f " will usually be abbreviated to "invariant under Uf".
LEMMA. Co'onsider n family oj' cztrves o(x,y) = c not identical with tlw
jamily o f path curves of the group generated by the infinitesimal trans-
fmmation Uj: 172e nhrves w = c are perm~cted among themselves hy every
transformation of tlze group i f and only $' Uw is n function of w.
First, if the curves w (rc, y) -=c are permuted by every transformation Ttof
the group, then to each pair of values of t and c must correspond a value
y ( t , c) such that
(1) 6 ) !TI, YI) = y ( t , c)

for all sets of solutions 2, y of w(z, y) = c. Then by (13) of 5 8,

Conversely, let U w -
which is a function of c and hence of w .
F(w). Since w = c is not a path curve, UO)is
not zero identically by Theorem 2 of 4 12. By (12) of 4 8,
304 L. E. DICXSON.

Since F(wl) is not zero identically, there is a region in which the int,egral

+
exists. Evidently I(o,) = t k, where k is a constant. This becoilles
+
I(w\ = k for t = 0. Hence I ( o , ) = f I(o),whose solved form is of
type (1). This proves the lemma.
The path curves ~ ( xy), = c are individually unaltered by the group
and we have Ut: 0 by Theorems 1 and 2 of % 12. This result and
the lemma together imply the followii~gconclusion:
THEOREM.An o~dinal-yrliferential equation of the first order having
the integral @(x, y) = c 2s invariant ~ m d e r (the group generrcted by) t11r
infi?zitesimcrl transforsnation Cyf) iJ. a?zcE only if Ccl, is n jir?lc~tion oj' a .
15. Finding an integrating factor. Let the equation

be invariant under the infinitesimal transformation

U@
UF
-
and let the path curses of the group generated by U f be not identical
with the integral curves 0 = c of the diiferential equation. Then by 8 14.
q ( @ ) 0. We can find as follows a function F ( 0 ) such that
1:

Since F(@), as well as 0 , is an integral of (2), it is a solutioil of the

corresponding partial differential equation (S 1.3)

(4) P f zzz A -a+

f B - = 0af.

-
ax a?j

Hence
aF
A-+B-
aF
= 0, UF aF
5 -$q- aF 5 1.
ax aY ax aY
 DIFFERENTIAL EQUATIONS FROM THE GROUP STANDPOINT. 305

The common multiplier is not zero identically since A and B are not both
zero identically by (2). Hence

Since this fraction is an exact differential d F , it is customary to say that

(2) has the inte,qratiny fartor

THEOREM. If the differential eqztafion B d x - d d y = 0 i s i n v a ~ i a n f

~ t n d e rthe i~zjinitesimaltransformation Uf given by (3) and if the path
cltrves o f the groztp genwated by U f are not identical tcitlz the inte,yrnl
curves o f the diferential equation, tlzen ( 6 ) is a n integratin,. factol. o f the
equation.
The condition on the path curves may be replaced by $B - 7 A $ 0.
For, then U@ $ 0, since otherwise Q, would be a common solution of
UQ,= 0, PQ, = 0 , and, since the determinant B -q A of the
coefficients is not zero identically,

and Q, would reduce to a constant, contrary to hypothesis.

COROLLARY.If B d x -A d y = 0 is invam'a.rtt under Uf, it hm the
integrating factor (5) provided the denominator is not zero identically.
16. Solution of simple problems.
PROBLEM 1 . Find an infinitesimal transformation which leaves invariant
the general linear differential equation

of the first order, and integrate it by use of the theorem of 5 15.

The corresponding "abridged" equation

R(x)z = 0

-
z'+
has the integral
t( x )
-SR&

e
 +
Then if yi,) i, any integral oi (6), ?j n z is evidently an integral when
(r i. any constant. I n other ~ ~ o r d (6)
s , is illvariant under the group of
tran~iorn~ation~
J1 = I . y, = y + ( / ? .

Hy 126) u t 2 9, tllc gron11 i. generated by the infiniresinla1 transformation

Hence -1 'z is nn integrating factor of

(G1) (Q- R y ) d x - d ! j = 0.
The produrt obtained is therefore an exact differential, d m . Hence

Solring 0 = I. for y, me get

PROBLEN 2. Find every curve such that the radius vector to any point P
on it make* the same angle 8 with the tangent at P that it makes with
the T-axis.
Let L. and be the ~,ectangularcotirdinates of P.

(I
-
Then

tall Y

I/

(7 7
-.
= Y
I

= tall 2 19 =
t a n s = -.

2 tan8
1-tan2@
'1 y
d n:

--
T

- 2 J"?/
=2 8.

xe- Y2 ' 0
1 B
/
,,"'

(7) 2 ~ , g d . r - ( x ~ - ~ ~ ) ( Z=
! / 0. Fig. 3.

It we nlagnif: our figure uniformly from 0, we evidently obtain a new

curve of the desired family of curves. In other words, each of the trans-
tormation%
(8) = ax. ?/I = Q Y
 DIFFERENTIAL EQUATIOMS FROM THE GROUP STANDPOINT. 307

merely permutes the curves of the family and hence leaves invariant the
differential equation ( 7 ) of which they are the integral curves. This may
be seen also from the fact that ( 7 ) is homogeneous. B y (26) of 5 9, the
group of transformations (8) is generated by the infinitesimal transformation

Hence an integrating factor of ( 7 ) is the reciprocal of

~(2xy)-y(33~-y" = (GC9+y3)y.
Thus
22dx
-- (.r2- ys)d y
+
.ra yP +
( x 2 y2)y

is an exact differential, d o . Hence

a - - 23:
-
as
A

x3+y2,
0 = log (x' + ye) + my),

Thus one integral is log (x8+ ya)-log y = const., so that another is

+
sa y' + cy. The required curves are therefore the circles tangent to
the x-axls a t the origin.
The work becomes simpler if we use polar co6rdinates. Then

tan OPB = - - tang, t a n @ . d e - - . e d 6 = 0.

de

Since this is invariant under e, = a e , 8, = 6, and hence under

an integrating factor is the reciprocal of q tan 8. The effect is to separate

the variables :

de
--- do - -0, loge-logsino=C, e=csin6.
e tan 8
308 L. E. DICKSOX.

P R O B L3.E ~ FI ind ever1 curve such that the radiu* vector tn any point P
on it makes a constant angle with the tangent a t P.
Denote the tangent of the angle by l i k . Then

Since the curves are evidently merely permuted b~ the magnification (8),
we find from (9)that 1 q is an integrating factor. The latter follows also
hj- n i n g the infinitesimal rotation a f 86 in the polar coordinates e, 8.
This generates a group n-hich merely permutes the curves. Multiplicatioii
b? 1 q separates the variables and we get the logarithmic spirals e = cde.
I i . Infinitude of groups leaving invariant a differential equation.
In the last problem we noticed that a certain differential equation is in-
variant under t n o grollpb. X7e shall n o v prose that any differential equation
-
5 d r -4dy = 0 is insarianr under infinitely many groups. There exists
ail integrating factor 1 ,1. mhence

Clioc~se any pair of functions E(r, y), q(z,y) such that

this choice being l~ossiblein infinitely many w a p . Then

where C'j' is the infinitesimai transformation (3). Since UCD r 1, the

theorem of 14 shows that our differential equation is invariant under t7J
THEORESI.For each of the inJinitlrde oj* p a i ~ s o f fz~nctions F and q
scitisfyiny (lo), Bdx - d d y = 0 i.s invariant under the infinitesimal
f~ansforntntion rf = ;ff , $ 4 fv.
18. Geometrical interpretation of Lie's integrating factor. Let

(11) Bdx-Ady =0
 DIFFERENTIAL EQUATIONS FROM THE GROUP STANDPOINT. 309

be invariant under the infinitesimal tl.ansformatioa

(12) af
LYE z-!--+q-.-. a)"
- ax a!/
Consider the three points (x, y),
( x + d , y + B ) , (x+Lr, y4-7). The
second lies on the tangent to the /x.y
integral curve through (x, y) of (ll),
since its slope is BIA. he third Fig. 4.
point lies on the tangent to the path
curve through (n., y) of the group generated by C:f; since its slope is
q l E by Theorem 2 of 5 12. The area of the parallelogram of which these
three points are consecutive corners is the absolute raltle of the determinant

whose reciprocal is the integrating factor ( 5 ) of 5 16.

The area of the parallelogram is also the product of the length I/da+ BZ
of one side bv the projection on the normal to the initial integral curve
of the length of the adjacent side. By the differential equations

of the group, the element of arc along the path curve through (x, y ) is

I/llz8+ d y e = m.
df.

Let d n denote the projection of this element of arc on the same normal.
Then the projection of the side of length is d n l d t , so that the
area of the parallelogram is w. d n l d t,
THEOREM.I f the di$ere?btiaE equation ( 1 1 ) is incariant z~ncler tlze in-
jinitesimd transformation (12), i t has the integrating factor
310 L. E. DICKSON.

19. Parallel curves. If on each normal to a given curve Co a segment

of constant length t is laid off, their end points form a new curve Ct.
When t varies, we obtain a family of curves called parallel crtrves.
An arbitrarily chosen point P: (x, y) determines uniquely a curve Ck of
the family. Let N be the normal to Co which passes through P. The
intersection of i17 with Ck+t is a point PI whose coardinates x ~ y, , are
functions of x, y and t. These functions define a transformation Tt which
replaces P by PI. Similarly, the transformation Tt. replaces PI by the
point P2 of intersection of N w i t h Ck+t+t.. Since the product Tt Tt.replaces
P by P,, we have Tt Tt = Tt+t,. Hence all the transformations TLform
a group whose path curves are the normals to Co. The element of arc d7t
on any normal is dt. Hence we may suppress d n l d t from the integrating
factor (13).
TEEOREM. A differential equa timz B d x - d y = 0 whose integral curzjes
are parallel mnes has th integrating factor .-/I
By the involutes of a curve C are meant the curves which are orthogonal
to the tangents of C. It is proved in the calculus that the various in-
volutes of C form a system of parallel curves. Hence, the preceding theorem
gives an integrating factor of the differential equation of the involutes.
This equation is readily obtained.
When C is the circle x 2 + yZ= 1, the equation of the tangent having the slope p ia

Hence this gives the slope of a tangent to an involute of C. Choosing, for example, the
upper sign and equating the fraction to dyldx, we get the following differential equation
of the involutes:
(XY V m ) d r + iy2-l)dy = 0.
Here A2 + B2 is seen to be a perfect square and an integrating factor is the reciprocal
of z + y V X+~y" 1. The usual method of integration yields the equation of the involutes :

20. Isothermal curves. Let w (2, y) = c be the integral curves of

(14) B d x - A d y = 0.
They are cut orthogonally by the integral curves L(x, y) = c' of

(1 5 ) A d z - + B d y = 0.
 DIFFERENTIAL EQUATIONS FROM TEE GROUP STANDPOINT. 31 1

Through an arbitrarily chosen point P: (x,y) passes a unique integral

cun7e o ( 2 , ~= ) k of (14) and a unique one S of (15). The intersection
of S with o(7.y) = k t f is a point P, whose corirdinates /,.y1 are
functions of J - , y, f. These functions define a transformation Tt whicll
replaces P by PI.Similarly Tt,replaces P, by the point P2nf intersection
of S with o (.T, y) = k f f + Then Tt Tt, and T+,(,
f'. a re identical since
each replaces P by P2. Hence all the Tt form a group whose path curves
are the integral curves of (15).
Interchanging the r8les of o and I., me get a second group of tran-
formations Ti whose path curves are the integral curves of (11).
If the ratio d n l d f of the element of arc dn of a path curre to d t is
the same function of I. and y in both cases. me call either family of cnrre>
a system of isothermal* curves. Since also the sum of the squares of the
+
coefficients in either (14)or (15) is d a B g , it fo11o~~s from the theorem
in 5 18 that the two differential equations have a common integrating
factor M7e proceed to show that *If can be found by quadratuses (thc
integration of a function of a single rariable being called a quadrature).
Since the product of (15) by 24 is an exact differential, say d y ,

each member being a value of a 3 y / a za y. Hence (16)is the condition that

IJIddx+ ,lfBcZy be an exact differential.
Performing the differentiations in (16) and dividing by Jf, we get

To derive the corresponding resnlt for (14), we have only to replace d by B

and B by -A; hence

*Thus a family of isothermal curves, together with the curves cutting them orthogonally.
divide the plane into i'infinitesimal aquares". For example, all concentric circles form a
family of isothermal curves; they are cut orthogonally by the radii.
312 L. E. DICKYON.

Sol~ingthe last two conditions as linear equations, we get

-- aal l tan-' -
a 1 0 g ~--- B a log (Ae
-4- + BZ),
(17)
ax d ax

8logM --a tan-1 --

B a
aY
-
ax 4-log
A ay
+
(-4% BY).

Xultiplying (17) by dx and (18) by d y and adding, we get the value

of d log N,so that logd4 can be found by quadratures and then M is ob-
tained. The condition that the value of d logM be an exact differential
is (as shown above) that the partial derivative of (17) with respect to y
be identical with that of (18) with respect to x, and hence is

Conversely, when condition (19) is satisfied, equations (14) and (15) have
a common integrating factor and (13) shows that d n l d t is the same at
any point for the two families. Hence the integral curves of (14) are
isothermal curves.
THEOREM. The integral curves of Bdx- A d y = 0 are isotlzermal if nnct
only if condition (19) is satisJied.
For the concentric circles with center a t the origin, whose differential equation is evi-
dently s d x + y d y = 0, we have B / A = - s l y , and (19) is seen to be satisfied. From
(17) and ( l a ) , we readily get M = l / ( ; c 2 + y2), which is a common integrating factor of
the differential equation of the circles and that of the radii, y d x - s d y = 0.
+ -
Again, the family of circles tangent to the x-axis a t the origin, x2 g2 cy = 0, has
the differential equation 2 zy d z - (x2- y2)d y = 0. Thus

Hence by the preceding eximple, condition (19) is here satisfied. The differential equation
+
of the family of curve8 orthogonal to the above circles is (xZ- y3 dx 2 z y d y = 0, whose
 DIFFERENTIAL EQUATIONS FROM THE GROUP STANDPOINT. 313
+
integral curves are the circles x2 g2- cx = 0 tangent to the y-axis at the origin. This
is evident since the second differential equation can be derived from the first by inter-
changing z and y ; or we may employ the common integrating factor
1
M =
(z" yY2 '
which is easily derived from (17) and (18).
Readers interested in the differential geometry of surfaces will find
applications of the above and similar principles to minimal, isothermal,
or asymptotic lines, and lines of curvature of surfaces, in Lie-Scheffers,
Differentialgleichungen, pp. 160-187 (or the resume in Cohen's Lie Theory
of One-Parameter Groups, pp. 78- 82).
21. Commutator. Given two linear partial differential expressions

we define the comrnzitator (alternant or Klammerausdruck) of Uf with Pf to be

(UP)f -- U(Pf)-P(Uf),
and shall prove that, after cancellations of second derivatives, it reduces
to a third linear partial differential expression. We have

If each time we differentiate the second factor of the products, we get the terms

which cancel in pairs. Next, if we differentiate the first fa,ctor of the

products, we get
 L. E. DICKSON.

which may be written in the form

Hence, the commutator (CP)f is found by the follo~ving simple rule.

The first coefficient (that of aflaa) of the commutator is found by applying
the operator Uf to the first coefficient -4 of Pf and subtracting the result
of applying the operator Pf to the first coefficient of Cf, and similarly
for the second coefficient.
22. A second criterion for the invariance of a differential
equation under an infinitesimal transformation. Our former criterion
(§ 14) depended upon a knowledge of an integral of the equation. Since
an integral is not known in advance, but is just what we are seeking,
that criterion was mainly of theoretical value. We shall now establish
another criterion which does not assume the knowledge of an integral and
can be tested a t once on any given equation.
Let 0 be an integral of the differential equation and let the corresponding
partial differential equation be

so that P(D = 0.
Assume that the equation is invariant under Uf defined bj- (20). Then
LT(D is a function q ( 0 ) of (D by 8 14. Hence

-
the former are proportional to those of the latter and (UP)f
where a is some function of x and y.
Conversely, ( U P )f aPf implies
-
so that 0 is a solution of the linear partial differential equation obtained
by equating (22) to zero, as well as of (23). Hence the coefficients of
aPf)
 DIFFERENTIAL EQUATIONS FROM THE GROUP STANDPOINT. 315
whence I/'@, being a solution of P f = 0, is a function of 0, Hence
(5 14), our initial differential equation is invariant under U f .
THEOREL* An ordina9.y dijerentiab equation is invariant ztncler fhe in-
jnifesimal transfol-ntation U f if and only if ( U P )f = uPf, whwe Pj'= 0
is /lie correspondiny yap-tial ( l i f e ~ e n t i a lequation.
For example, consider the linear dserential equation

The corresponding partial differential equation is

W e seek a function r (x) such that Pf = 0 is inv~riant under the infinitesimal trans-
formation of the special type
af
Of E z(x) -
a !/
e~nployedin S 16. Since
(UP)f r (-zR-2') af
-
aY
lacks 8 f l a x , it will be a multiple of P f only if it is zero identically, whence z is an
integral of the abridged equation (5 16).
23. Group of extended transformations. The point transfoimlation

transforms any direction y' = d y l d x into the new direction

The combination of a point (x, y) and a straight line (of slope 2 ' ) through
it is called a lineal element. The three equations (24) and (25) define a
transformation T"a of lineal elements which is called the (first) extended
transfwmation on the three variables z, y: y'.
These extended transformations form a group if the transformations (24)
form a group. For, if Ta, TI, and Tc are identical transformations (24) and
hence take any curve Cl into the same curye Ce, the extended transfor-
mations Ti, Ti and T,'both take the points and slopes of the curve CT1
into the points and slopes of Ce and hence are identical transformations.
*Another proof of this theorem is given a t the end of 4 28.
316 L. E. DICKSON.

Thus the product of any two extended transformations TA and Ti is an

extended transformation T,'.Kext, if (24) is the identity transformation
when u = a ~ then
, x r Q, (r,y, ao), y r 9 (2, y, Q,), whence

(26) az = 1, Q, = 0, VJZ = 0, = 1, when a = ao.

Thus (25) reduces to yi = yf for a = G,SO that T& is the identity trans-
formation on x, y, y'. Finally, if Ta takes any curve C into the curve C,,
the inverse transformation Ta,takes Cl into C, whence Tk T& leaves in-
variant each point and each slope of C and is therefore the identity trans-
formation, so that TL has the inverse To',.
By (26) of 8 9, the i~lfinitesimaltransformation of the group (24) is

To find the infinitesinial transformation of the extended group, we differ-

entiate equations (24) and (23) with respect to a and then take ct = a,,.
This was done above for (24). Sext, denote the fraction (25) by h7/D,
Then, by (26),

when a = ao. Write q' for the value of the derivative of hTID when
ci = no. Then

The symbol of the infinitesimal transformation of the extended group is

therefore

24. Invariants. A function f(x, y) is said to be invariant under a trans-

format#ion on x and y if it remains unchanged by the transformation. Thus
f(x, y) is invariant under a group of transformations with the parameter a
if and only iff (xl, yl) =f (x, y) for all values of x, y,a. Then df ( q , yl)ld a E0.
Conversely, this implies that f (xl, y,) is independent of a and hence equal
to its value f(x, y) for a = ao, whence x, = x, y, = y. Hence by 8 8,
 DIFFERENTIAL EQUATIONS FROM THE GROUP STANDPOINT. 317
j'(.r,y) is ittcariant ~ n d e r fhe p o u p yenerafed by tlze infinitesiw~al trans-
jornznfion Uj'if nncl only fl lT is identically zero in x and y.
Por exa~nple, r z + y Z is invariant under every rotation about the origin 0 since the
distance v-
enimnl rotation is
of the point (2,y) from 0 is unchanged by the rotation. The intinit-

+ y Z is invariant under the group of

niid r 2 rotations since

25. Differential invariants. A function F(x, y, y') which actually in-

volves the dei-ivatire IJ' is called n difereiztial ilsva?.innt of the first order
of :Lgroul) of point transformations (24) if, when F is regarded as a function
of three independent variables, it is invariant under the group of the first
extended transformations defined by (24) and (25). This will be the case
if and only if C ' F is identically zero in J, y, y', where U ' f ' is the in-
finitesimal transforniation (28) of the extended group. In fact, the discussion
of invariants given in $ 2 4 holds true for three or more independent variables
;IS well ;is for two.
For example, let Uf' be the infinitesimal transformation

By (2'i), - -
of the group of magnifications from the origin 0 (similarity transformations).
0,whence U ' f U j : Hence any function of y' is invariant
under U ' f and is therefore a differential invariant of U f . This is obvious
geometrically since the slope y' of any straight line through the origin 0
is unaltered by a lnagnification from 0.
26. Determination of all differential invariants of the first order
of Uf. W e desire the solutions, involving y', of

The corresponding system of ordinary differential equations is

-dx d y-
=-= dy'
5 T 9' '
318 L. E. DICKSON.

The equation d z l S = dylq, in x and y only, has an integral u(x, y) = c,

Solve the latter for y in terms of x and c, and substitute the value of y
in dxlt' = dg'lq'. Employing the expression (37) for v', we get

where p, q, T are functions of x and c. This Ricatti differential equation

of the h s t order has an integral 0 (x, y', c) = k, where k is an arbitrary
constant. Replacing c b ~ zc.- (x, y), we obtain a second integral v(x, y, y') = k
of the system (31).
By differentiation of v = k we have

Replacing dx, dy, dy' by the expressions 5, q, which are proportional to

them by (31), we get

which shows that v is a solution of (30). Similarly, u is a solution. Also

an arbitrary function ~ ( z L
v) , of PC and is a solution since

Furthermore, every solution of (30) is of the form q ( u , v). In fact, the

solutions u and a are independent, since v involves y' while u is free of y'.
If there were a solution w independent of u and v, then x, y, y' could be
expressed as functions of u, v, w and hence would be solutions of (30),
whence 5, q, would be identically zero, contrav to hypothesis.*
THEOREM.All diferential invariants of the .first mder of the group
generated by Uf are $~nctions involvin,g y' of two independent solutions
u(x, y) and v(x, y, y') of U'f = 0 which may be foztnd by integrating two
ordinary di$erential equations of the first mder in two variables.
* To give another proof, introduce u, v, w as new variables. By $ 29, the coefficients
of the new partial differential equation are identically zero. Returning to the initial
variables, we conclude that the coefficient8 of (30) are all identically zero.
 DIFFERENTIAL EQUATIONS FROM THE GROUP STAYDPOINT. 319

For example, when Uf is given by (29), we may evidently take u = y l x , v = y'. Hence
every differential invariant is a function of ylx and yl.
Again, to find all differential invariants of

which is the infinitesimal rotation about the origin if n = -1, we note that I]'=1 - n y f 2
by (271, so that
+ af
ax +
af
~ t Ef n y - x -8.t (l-ny' 2) -7.
(33) ay ay .
The corresponding system of ordinary differential equations is

An evident integral is u = z2-n yZ. To find a second integral v , denote each of the
three equal fractions by f. Then
xdy-ydx - f =
dx = nyf, d y = zf, -- -
x2 n y 2 - - dy'
1 -nyl I '
Write n = -c2. Then an integral is tan e ylx -tan-' cy', whose tangent is the
product of c by
2) Y-ZY'
x-nyy"

27. Invariant equations. We shall call o(x, y) = 0 invariant under

the group G generated by Uf if o(xl, y,) = 0 for all values of t and all
values of x,y which satisfy w (x, y) = 0. Then the derivative of o (xl, y,)
with respect to t must be zero for those values. Hence by (13) of 5 8,
Uo)(z,y) must be zero for all values of x, y satisfying w(x, y) = 0.
Conversely, let this condition hold. In case 5 = 7 = 0 for every point
(2,y ) on the curve o --- 0, each point and hence the curve is invariant
under Q (8 11). We shall exclude the singular case in which the derivatives
ox and u, are both zero at every point on w = 0. Let (x, y) be any point
on the curve for which those derivatives are not both zero, and also 5
Since
0. *
Uw = wxS+wyq = 0

at (x, y), we have o, # 0. Hence for the slope d y l d x of the curve o = 0

at (x, y), we have

But 7i/5 is the slope of a path curve of the group by Theorem 2 of 5 12.
Since the differential equation t l y l d x = ri/E has a unique integral curve
320 L. E. DICKSON.

passing through (2, y), and w = 0 is one solution, it follows that o = 0

is a path curve or an aggregate of path curves and curves each of whose
points is invaria~~t under Q. Since each path curve is invariant (8 12),
w = 0 is an invariant curve.
THEOREM.A curve w (x, y) = 0 is invariant wnder the group generated
by U f i f and only if Uw is zero for all valz~esof x and y which satisfy w = 0.
A like result holds for surfaces (II(x, y, z) = 0 under transformations on x, y, z.
To prove the last part of the theorem, suppose that

for all values of x, y, z which satis@ w = 0. Let (x!y, z) be any point

+
on the surface w = 0 for which g, 7, E; are not all zero, say t 0. Since
WZ, w,, w, are proportional to the direction cosines of the normal to the

surface w = 0 at (x, y, zj, it follows from (34) that E, q, E; are proportional

to the direction cosines of a tangent to the surface a t (x, y, 2). But the
direction cosines of the tangent to a path curve of the group generated
by Uf are proportional to d x , dy, dz and hence to E, 7, t. Thus at each
non-invariant point of the surface w = 0 the tangent to the path curve
is a tangent to the surface. It may be shown as follows that the path
curve lies on the surface. We may take w = 0 in the solved form z = z ( x , y).
From the differential equation

we ge,t y = y (x,c), where c is an arbitrary constant. From this and

z = z (z, y (2, c)), we get

since (34) becomes 52, +qzv-5 = 0 for w G Z(X,y) -2. Hence the unique
(path) c m e through (x?y, z) for which clx: d y : d z = f : q : I; is on the
surface w = 0. Since w = 0 is the locus of an infinitude of path curves,
it is invariant under the group.
28. A third criterion for the invariance of a differential equation
under Uf. Under a point transformation (24) and the induced trans-
formation (25) on y', a differential equation F(x, y, y') = 0 is transformed
into an equation Fl (xl, yl, y;) = 0, and the integral curves @(x, y) = c
of the former go into certain curves CD, (xl, y,) = c. Since any set of
 DIFFERENTIAL EQUATIONS FROM THE GROUP STANDPOINT. 321
solutions x, y, y' of F = 0 goes into a set of solutions X I , yl, yi of FI = 0 ,
while any point ( x , y) and slope y' of the curve Q, = c go into a point
(x17yl) and slope y! of Ol = c, it is evident t,hat the transformed curves @I = c
are integral curves of Fl= 0. Hence the integral curves of a di$erential
equation are transformecl By any point transformation into the integral
curves of the transfornzed equation.
In particular, the differential equation F = 0 is invariant under a group
of point transformations if and only if the ordinary equation F = 0 in
three independent variables x, y, y' is invariant under the group of extended
transformations defined by (24) and (25). The condition for the latter is
given by the theorem in 5 27.
THEOREM. A dzffwential equation F(x, y, y') = 0 is invaviant under the
group of point transformations generated by an inftnit~dmaktransforrr~ation
Uf i f and only i f U ' f vanishes for a22 sets of values x, y, y' satzijkzng
F = 0, where U ' f is the Jirst extension (28) of U f .
-
For example, let Cf be the infinitesimal rotation given by (32) for n = 1, so that
U'f is (33). The lines y/x = c are merely permuted by a rotation about the origin.
They are the integral curves of the d8erential equation xy'- y = 0, which is therefore
invariant under Uf. This fact also follows from the present theorem since

Again, the differential equation of the tangents to x2+ yZ = 1 is

which is invariant under tho same Uf since U' 3' = 2 y' 3'.
Let us apply the preceding theorem to the case in which F has the
solved form F = Ay' -B = 0, where A and B are functions of x and y
alone. We have

U'f ZE U,f + rl' aaYf

7,U'F - y' UA- U B $ t/A.

We are to employ all set's of values x, y, y'

For y' = BIA, the expression (27) for
-
Bl.4 for which
becomes
F = 0,

where Pf - Afx + Bf,;

1
-(UA-
and the quotient of U ' F by B becomes

1
PF) -- (uB-Pq),
A B
322 L. E. DICKSON.

which, by the theorem, shall be zero identically in x and y. Then by (22))

the coefficients of ( L T )f are proportional to those of P f . This gives a new
proof of the theorem of § 22 which states that A d y - B d z = 0 is invariant
under Uf if and only if (UP)f -= a P f , where o is a function of x
and y alone.
29. Introduction of new variables in a linear partial differential
expression. We shall express

.
in terms of new variables y,, . ., y, which are independent functions of
.
XI,. . ., x,,. Since we may express f (x,, . ., xn) as a function of y,, ..
., y,,
we have
af=xay,+...+
- 8.f ay,
-
a ~ i ayl 8xi a y n a&'

Multiplying by Ai and summing for i = 1, .. ., n, we get

.
These sums are equal to Py,, . ., P y n respectively.
THEOREM.If in a n y linear partial di$erentiaZ expression P f i n n
variables we introduce n new independent variables y,, ..
., yn, we obtain

wlzere [ P y i ] denotes the fztnctio~z obtained by q r e s s i n g P y i in t e r n

of Y I : . . ., Yn.
For example, let Pf and the new variables p, 8 be

Then Pp = 0,PO = 1, so that the infinitesimal rotation Pf about the origin becomes
afja8 when expressed in polar coordinates.
30. Determination of all differential equations invariant under
a given infinitesimal transformation U ' By 5 28, F(x, y, y') = 0
is invariant under Uf if and only if U'F vanishes for all sets of values
2, y, y' satisfying F = 0. In particular, F = 0 is invariant if. U'F is
 DIFFERENTIAL EQUATIONS FROM THE GROUP STANDPOINT. 323
identically zero in x, y, y'; then F is a differential invariant of U f (5 25)
and is a function of two solutions u (x, y) and v (x, y, y') of U'f = 0 (5 26).
Conversely, if we equate to zero any function, actually involving v, of u
and v, we obtain a differential equation invariant under Uf, since u, v
and any function of them are invariant.
Are there dserential equations F = 0 invariant under U f which are
not expressible in the form q(z(,v) = O? I t seems plausible that U ' F
may vanish for all sets of solutions of F = 0 without being identically
zero. For if

V F = x is not identically zero and yet T% vanishes when F = 0.

In answering our question we may assume without loss of generality
that u(x, y) is not free of y (otherwise, we repeat the following discussion
with x and y interchanged and 5 and g interchanged). This assumption
and the fact that u is a solution of U f = 0 imply that P is not identically
zero. Then x, u, v are independent functions of x, y, y' and may be
introduced as new variables; by 3 29, U'f becomes

where R denotes the function obtained by expressing U'x = F in terms

of the new variables. Let F = 0 become b(x, u, v) = 0 when expressed
in terms of the new variables. If x does not actually occur in o, we have
an equation in u and v mentioned above. Suppose next that x occurs in
a = 0, which may therefore be written in the solved form x- @ (zc, v) = 0.
Since it is invariant under R 8flax by hypothesis, R is zero for all sets
of solutions of x- 10 = 0. Returning to the initial variables, we conclude
that 5 is zero for all sets of solutions of F = 0. But the differential
equation F = 0 involves y' and serves to determine y' as a function
f (x, y) of x and y. Hence 5 is zero for all sets of values x, y, y' =f (x, y).
Since y' does not occur in 5, 5: is zero for all values of x, y, contrary to
the fact that F is not zero identically in x, y.
THEOREM.A d i ' e n t i a l equation of the Jirst order is invariant under
the group generated by the injniteximal transformation Uf if and only zf
it can be expressed as an ordinay equation between two independent solutions
zc and v of U'f = 0.
Two examples were discussed at the end of 5 26. The first appears as
the case s = 1, r = - 1 of line 2 of the following table, while the
324 L. E. DICKSON.

second appears in line 6. The result (5 16) for a linear differential equation
appears as the case s = 0 of line 3. The further entries in the table
were obtained by starting with infinitesimal transformatior~sso chosen that
we can perform the integrations necessaq- to compute the invariants 21 and v.

TABLEOF DIFFEREXTIAL EQUATIOXS INVARIBST UNDER THE ACCOhIPASYING

INFISITESIhlAL TRANSFORMATIOXS.

jtotatio?~~: Y=-, af
ax
(1"- 81'
aY ' n, T, s constants.

Y ' = f(9-z+sy), sp-?.q.

xy'=yf(xrySj, sxp-ryq.
$8-1) dl
g 1 - t - R ( 4 y = Q(x)@, Y"q.

2:
-
=
y -xyl =f(x3--92yS) .: x-nYy1
1 -nv2
1
x-nyy

.. 1 - c %,/noEz ( x if ix y- n)y(y l' I G y ' ) = h ( x g - n y ' ) ,

Further types may be derived from these by interchanging x with y

and hence replacing y' by l i y l . All further types listed in the books by
Cohen and Page are special cases of the foregoing. This table senTes as
a key to the integration of differential equations of the first order.
 DIFFERENTIAL EQUATIONS FROM THE GROUP STANDPOINT. 326

Solution of one or more linear partial differential equations

of the first order.
After presenting standard methods of solving a single linear partial
differential equation in three variables, we shall give the simpler methods
available when the equation is invariant under one or two linown infini-
tesimal transformations. The latter methods are based on the classic
theory of complete systems, which is developed in full detail for the
typical case of two equations in three variables (the only case required
in this paper except in the discussion of covariailts in S 55). Ia particular,
this case is sufficient for the important applications in chapter IT to
ordinary differential equations of the second order.
31. Existence of solutions. To the equation

in which A, B, C are functions of x, y: z, we make correspond the system

of ordinary differential equations

If ~ ( xy,, z) = const. is any integral of (P), it mas readily verified in

(5 26 that v is a solution of (1). Also, if tt and v are tn.0 independent
solutions, all functions of u and v are solutions and exhaust the solutions
of (1). The relation between the equations (1) and (2) in three variables
is analogous to that established in 5 13 for equations in two variables.
F o r the case in which A and B are functions of x and y only, it was
proved in 5 26 that there exist two independent integrals z( = c, 1: = k
of (2). For the general case in which A and B, as well as C, are any
functions of x, y, z, the same conclusion holds.*
Picard, Trait6 d'analyse, 11, 1893, p. 298. To give a plsusible argument (not a proof),
note that, at an arbitrary point P: (xo, yo, 2,) in space, equations (2) determine a unique
direction or straight line whose direction cosines are proportional to A (xo, po, z0), B, C.
Consider a point P1:(x,,y l , 2.,) in this line a t an infinitesimal distance from Po. Equations
(2) determine a unique direction a t P, and hence deternline a third point P2 at an infi-
nitesimal distance from PI and in the latter direction from it. In this manner we obtain
a curve in space. By varying the initial point Po, we obtain in all a doubly infinite
system of integral curves of (2). The general one of these curves is determiued by tmo
equations involving x, y, z and two arbitrary constants c and k, whose solved forms are
(x, g, Z) = C, v = k.
326 L. E. DICKSON.

9 n y linear partial differential equation of the first order in n variables has

n - 1 independent solutions. Any function of them is a solution and every
solution is a function of them.
32. Three equations in three variables. Consider

..
in which A, . , I are functions of x, y, z. Suppose they have a common
solution a (x, y, z ) not a constant. Then

are not all zero. When 0 is substituted for f, equations (3) become three
linear homogelieous equations in the three unknowns (4). Since the latter
are not all zero, the determinant
ABC
(5) d = DEF
Q H I ,
of their coefficients is identically zero.

-
Then there exist three functions 1, nt, FZ of x, y, z not all identically
zero, such that
(6) lPf+mUf+nVf 0,

identically in x, y, z for all functioils f,and the three equations (3) are
said to be linearly dependent. For, if the minors

of the elements of the f i s t column in (5) are not all identically zero, we
have (6) for 2 = a, m = -d, n = g. But if all nine minors of (5) are
 DIFFERENTIAL EQUATIONS FROM THE GROUP STANDPOINT.

identically zero and if I 0, for example, we see from a

where b is the minor of B, that
- -0 and b
327
0,

so that we have a linear dependence (6).

THEOREM 1. I f equations (3) have a common solution w7iich is not a
constant, they are linearly dependent.
The problem of solving three linearly independent equations (3) is there-
fore trivial, since their only common solution is a constant. If the equations
are dependent, so that (6) holds, where for example 1 is not identically
zero, every common solution of Uf = 0 and Vf= 0 is a solution of
Pf = 0, which may therefore be discarded. If four or more equations
in three variables have a common solution which is not a constant, all but
two of the equations are consequences of those two and may be discarded.
33. Complete system of two equations. In the commutator

of P f with Uf, the second derivatives are seen to cancel as in 5 21.

Whence if we employ the notations (3), we readily obtain

so that the rule for forming the commutator is the same as that in 5 21.
Let P f = 0 and U f = 0 be linearly independent and have a common
solution Q, not a constant. Then, by (7), Q, is also a solution of (PC:) f = 0.
Hence, by Theorem 1, Pf = 0, U f = 0, ( P U ) f = 0 are linearly dependent.
In the linear relation between them, the coefficient of ( P U ) f is not zero
identically (since otherwise P f and LTf ~vouldbe linearly dependent) and
may be divided out of the relation. This proves the following result.
THEOREM 2. I f P f = 0 and Uf = 0 are linearly independent and I?nl;e
a common solutimt whiclt i s not a constant, t h t

identically in x,y, z a n d for every fztnction f.

Two linearly independent equations P f = 0 and 1Jf = 0 whose commutator
is expressible linearly to terms of them, as in (9), are said to form a
complete system. Hence Theorem 2 may be stated in the following form.
328 L. E. DICKSON.

THEOREM 3. I f two linear partial difertxtial epzcations of thefirst order

are linearly independent and have a common solution which is not a constant,
they form a complete system.
In order to prove the converse of this theorem, we first show that if
we replace a complete system Pf == 0, U f = 0 by an equivalent system

in which a,. . . : d are functions of x,y, z, we obtain a new complete system

R f = 0 , Sf=O. By definition, (RS) f = R ( S f ) - S ( R f ) , whence

In computing each of the four terms Pf), etc., we must apply a linear
differential operator B or S to a product of two functions. Hence we
operate on each of the two factors in turn as in the dzerentiation of
a product. First, if we operate each time on the Greek letters, we get

Next, if we operate on the Roman letters, we get

Inserting the expressions (10) for Rf and S f , we get eight terms which,
after cancellation of four in pairs, may be combined by (7) into

Hence ( R S )f is equal to the sum of the expressions (11) and (12).

We assume that Pf = 0 and U f = 0 form a complete system, so
that (9) holds. Then (12), and hence also ( R S ) f , is a linear function of
P f and U f . The latter are linear functions of Rf and Sf, as seen by
solving relations (10). Hence

Hence if P f = 0 and U f = 0 farm a complete system, any two linearly

independent linear cmbinations (10) of t h m also form a complete system.
 -
DIFFERENTIAL EQUATIONS FROM THE GROUP STANDPOINT. 329
Our next step is to prove that we can choose R f and S f in (10) so
that ( R S )f 0. Since P f = 0 and U f = 0, given by (3), are' linearly
independent, not all the three %rowed determinants of their coefficients are
identically zero (since otherwise A, B, C would be proportional to D, E, F).
Let, for example, d = -4 E - B D 0. Choose
af af

-
1 1
R f - (EPf-BUf)=-+r%, 1.1-(EC-BF),
a ax a
1
Sj'= - ( - D P f + d U F ) =
af
-+ s
af 1
-(AF-DC).
a ay
S-,
a2 a
Then
(RS)f (Rs-Sr) - aj'
az '
Comparing this with (13), we see that I =- 0, p 0. Hence any complete
system is eqztivabnt to a ,JACOBIAi? SYSTEM whose commutator is identi-
cally zwo.
A Jacobian system has a common solution. For, let u(x,y,z) and v(x,y,z)
be two independent solutions of R f = 0. Then any function F(u, v) of
them is a solution. This F will be a solution also of S f = 0, if (5 29)

and recalling that Rzt 0, R v -

But ( R S )f = R ( Sf ) - S ( Rf ) r 0. Replacing f by ZL and v in turn,
0, we get R(Stc) r 0, R(Sv) r 0.
Thus Szc and Sv are solutions of R f = 0 and hence are functions g(zc, v)
and h(u, v) of u and v. Our equation (14) becomes

which is known (5 7) to have a solution F(u, v).

This common solution F of R f = 0 and S f = 0 is a common solution
of P f = 0 and U f = 0 in view of (10). This completes the proof of
the following converse of Theorem 3.
THEOREM 4. I f P f = 0 and U f = 0 form a complete system, they have
a common solution which is not a constant.
34. Complete system of r equations in n variables. The preceding
results may be extended readily by the same methods* to linear partial
*For a simple exposition of this classic theory, see L. Bianchi, Gruppi continui h i t i di
trasformazioni, 1918, 18-28; Goursat's Mathematical Analysis, vol. 2, part 2, p. 267 (English
transl. by Hedrick and Dunkel).
330 L. E. DICKSON.

differential equations Pif = 0 of the first order in l z variables. If n such

equations have a common solution which is not a constant, they are
linearly dependent. Hence consider s (s < n) linearly independent equations
Pl f = 0, . . ., P,f = 0. If they have a common solution which is not
a constant, each of the 1 / 2 s ( s - 1 ) equations

has this same solution. We therefore enlarge the given system of equations
by annexing seriatim such of the new equations as are linearly independent
of the former and those previously annexed. We form the commutators
of the enlarged system and again annex seriatim those which are independent
of the former and those previously annexed. This process must terminate
since we saw that not more than n - 1 independent equations have .a
common solution which is not a constant. Let the final system be

so that they are linearly independent, while the commutator of any pair
of them is a linear function of P l f , . . ., P,f. Such a system is called a
complete system. Hence all common solutions of any system are common
solutions of a certain complete system.
Our complete system (15) is equivalent to a Jacobian system* Q, f = 0, ...
. . ., Qrf = 0, all of whose commutators are identically zero. The latter
system, and hence also (15), has exactly n - r independent solutions. For,
by 5 31, Q1 f = 0 has n- 1 independent solutions 1 ~ 1 ,..., un-I. In view of
(Q1 Qe)f 0, Qe Z&is a function Q'i of ul, . .. .
, U n - l , Then F(u,, .. , h - 1 )
will be a solution also of Q, f = 0 if (5 29)

By 5 31, this equation in n - 1 variables has n - 2 independent solutions

s, ..'., vn-s. In view of (Q, Q3) f --
0 and (QgQ S ) f =-- 0,we see that Q8 vi
is a solution of both Q1 f -=0 and Q, f = 0 and hence is a function qi
of v1, . . ., ~ ~ - 2Thus
. @(q, . . ., vn-2) will be a solution of Q3 f = 0 if

* Obtained by solving (15) for r of the partial deriv~tivesin term of the remaining n-r.
 DIFFERENTIAL EQUATIONS FROM TEE GROUP STANDPOINT. 331
This equation in n -2 variables has n -3 independent solutions which,
being functions of vl, ...
, vn+ are solutions also of Ql f = 0 and Qaf = 0.
We have now taken three of the r similar steps in the proof (by induction) of
THEOREM 5. All common solutions of a complete system of r equu.tions in
n variables are functions of n-r independent common solutions.
35. Standard methods of solving a complete system of two
equations in three variables. One method employs two independent
solutions u and v of one of the equations, say U f = 0. By Theorem 4
of $ 33, there exists a function F(u, v ) which is a solution also of the other
equation Pf = 0, whence

If u itself is not the desired common solution, then P u 0 and

Since the partial derivatives of F(zg 2;) are functions of 14 and v, the remaining
quantity q in (16) must be a function of u and v. Hence (16) is an equation
in u and v only, and is equivalent ($ 13) to an ordinary differential equation
of the first order in two variables.
For example, the two equations

form a complete system since ( U P )f G 2 Pf. Evident solutions of U f = 0 are

u = y/x, v = z/x. Then

Thus (16) becomes

a
u - +f + v - af du = -
= 0, - ' , log U
= const.
au av u v v

Hence a common solution of (17) is u/v2 or xy/z2.

Another method reduces the solution of a complete system Pf = 0,
U f = 0, given by (3), to the finding of an integral f = const. of the
total differential equation
33% L. E. DICKSON.

' dx dy dz
A B C =0,
D E P

which follows kom the given equations and

To find an integral f = c of (18), note that for any chosen value of a

+
the plane z = x a y intersects the desired integral surfaces f = c in a
family of curves whose differential equation is f o ~ i l dby eliminating z and
d z from (18) by means of

and hence is of the form

For a arbitrary, let ty (x, y, a ) = c be an integral of the latter. Elimi-

nating a by means of z = x+ a y , we obtain the desired integral

of (18). The method requires mo&cation (as in the example) when a does
not actually occur in ty (x, y, a).
EXAHPLE. When the given equations are (17), we find that (18) is the product of
-8 (2' + Y') by
(20) ytdx+xzdy-2xydz = 0.
Inserting the values (19), we get

Since a drops out on removing the algebraic factor, the method fails and will fail when
any homogeneous equation is used for the planes. But we shall succeed if we employ
+
the equation y = s a, d y = d x , instead of (19). Now (20) becomes

Elimination of a gives x y/z2 = e.

 DIFFERENTIAL EQUATIONS FROM THE GROUP STANDPOINT. 333
36. Equation Pf = 0 determined by two solutions. Two in-
y, 2 ) and @,(cc, y, r ) of
dependent solutions @, (zt.:

determine uniquely the ratios of A, B, C and hence determine the equation

apart from a factor which is a function of z, y, a. For,

together with (21) require

whose expansion according to the elements of the first row is a linear

partial differential equation* which has the solutions 0, and 0, and is
uniquely determined by them.
37. Pf = 0 invariant under a transformation. By the simple
method of 9 29 we may express (21) in terms of new independent variables

(23) XI = l( x , y, a)! YI = P(X, Y, 21, .zl = Y(X, y, a);

denote the resulting differential expression by PI$

We shall say that Pf = 0 is invariant under the transformation (23) if

where A l , B 1 , Cl are the same functions of x,, y,, zl that A, B, C were

of x, y, a. Since @ i ( ~ y,, z ) is a solution of (21), cDi(x,, y,, 2 , ) is then
* Not identically zero. For, if the determinant (22) be identically zero, a well k ~ o w n
theorem states that f, @, are dependent functions. Since f is arbitrary, @, and @?
would be dependent contrary to hypothesis.
334 L. E. DICKSON.

a solution of PIf = 0. Since Plf = Pf in view of relations (23), the

latter solution becomes a solution of Pf = 0 when expressed in terms
of x, y, 2, and hence is a function of two independent solutions:

Conversely, relations (25) imply that Plf = 0 has the solutions cDi (x,, yl, 4,
so that (24) holds by 5 36.
Hence Pf = 0 is invariant under the transformation (23) and only
if its independent solutions @i satisfy relations (25).
Let an infinitesimal transformation Uf generate a group of trans-
formations (23) in which the coeficients are functions of a parameter t.
Then Pf = 0 is invariant under the group if and only if (25) are con-
sequences of (23), the coefficients of qi being functions of t. By. (15) of
Chapter I,

for every t sufficiently small, where @i denotes @<(x,y, 2).

If (25) hold, the coefficient of t in the expansion of its second member
must be equal to U@i by (26), whence*

Conversely, (27) imply that

and similarly that U[U(U@i)] is a function of (Dl, 0%.Then from (26)

we obtain relations of the type (25).
THEOREM 6. 8 9 1 equation Pf = 0 having the independent solutions @,
and (De is inziariant under (the group generated by) Uf if and only if U@,
and UQe are functions of and Og as i n (27).
This criterion for the invariance of Pf = 0 under Uf involves the
solutions Q)i which we desire to find. Hence we seek a criterion not in-

(UP) @i U(P@i)-P(UOi) -
volving the ai. First, let (27) hold. We employ the commutator

* These follow also horn (13) of Chapter I.

U(0)- P[Fi(@i, @z)] 0.
 DIFFERENTIAL EQUATIONS FROM THE GROUP STANDPOINT. 335

Hence ( U P )f = 0 has the same solutions mi as P f == 0, whence ($ 36)

Conversely, the latter implies that the solution 0% of P f = 0 is a solution

of ( U P )f = 0, whence P ( l i 0 i ) r 0, so that (27) follow.
THEOREM

-
7. P f = 0 is in~:(tria?zt zmde?.V f if and only i f the conzmzctator
( U P )f is the prodztct of P f by a jihnction of x, y, z.
In the example (17), (UP)f 2 P f . Hence P f = 0 is invariant under Uf.
38. Solution of equation P f = 0 invariant under Uf. First let
P f and Uf be linearly independent. Then, by (28), P f = 0 and U f = 0
form a complete system. Suppose we have found by one of the methods
in $ 3.5 a common solution 0 = 0 (x, y, z) which is not a constant. Then
a second solution can be found by quadratures as follows. By permuting
x, y, z if necessary, we may assume that z actually occurs in cD. Then
we may introduce the new variables x, y, 0 in place of x, y, z. By $ 29,
P f and U f become, when expressed in terms of x,y, 0,

since the coefficients of af l a 0 are P O - -

0, and U O 0. Here A,, B1,
El, ql are the functions A, B, 5, q expressed in terms of x, y, 0 . By (28),

so that Pl f = 0 is invariant under Ulj' and

is an exact differential (5 15), in whose integration by quadratures 0 plays

the r61e of a constant. Xote that the denominator is not zero identically
since otherwise PIf and Ulf would be linearly dependent, contrary to the
linear independence of P f and Uf.
EX AX PI,^. Let Pf and Uf be given by (17). W e found above the common solution
8 = z y lz2. Then

Hence Pf = 0 has the independent solutions x y / z 2 and x 2 + y 2 .

336 L. E. DICKSON.

Second, let Uf = vPf. We have

The case V E P gives (v P, P )f - -P v .Pf. Hence, by Theorem 7,

Pf = 0 is invariant under Uf = 71 Pf for every function v of x, y, z.
Since no new information is gained concerning a particular equation Pf = 0
from the fact that it is invariant under Uf r vPf, that infinitesimal
transformation should not be expected to aid in the solution of Pf = 0.
We now have Ulf -= v P l f , so that the denominator in (30) is identically
zero, and the fact that Plf = 0 is invariant under Ul f is of no aid in
its integration (cf. 5 15).
39. Jacobi's identity. If Uf, Vf, Pf are three linear partial differential
expressions of the f i s t order in any number of variables, then

(32)
Since

P)f

(UV)f -
+ (('OP) U)f 4-((P U )V)f
U(Vfl--V(U'f),

= 0.

((VP)U)f -
where U V Pf means U[V(Pf)]. Permuting the letters U, 77, P cyclically,
we get
VPUf-PVUf-CVPf+ UPVf,

By adding the three relations we get (32).

40. Equation invariant under two infinitesimal transformations
We shall first prove the existence of two infinitesimal t.ransformations Uf
and Vf each leaving invariant a given partial differential equation Pf= 0
in three variables (and such that Uf, Tf, Pf are linearly independent).
For, by $ 31, Pf = 0 has two independent solutions u(x, y, z) and v(x, y, z),
which are therefore independent with respect to two of the variables,
say x and y. In terms of the new variables xl = u, yl - - v, zl = z, the
equation Pf = 0 becomes (5 29)
 DIFFERENTIAL EQUATIONS FROM THE GROUP STANDPOINT. 337
By inspection, the latter is invariant under

These become the desired TJf and Vj'when we return to the initial variables.
We next prove the following fundamental result.
THEOREM 8. I f the linear partial d i f i e n t i a l eqtiation P f = 0 is invariant
ztnder two injnitesimal transformations V f and V'. i t i.s invariant also

-
zcndw tlieir commutator (UV)j".
By Theorem 7,
(33) (UP)f ePf, , (VP)f = cPf,
where e

Hence (32) gives

(( U V )P ) f
( P U )f -
and o are functions of x, y, z. Also,

(e P, V ) f -(GP,U)$
-(UP)$

BY (31),
(qP,V)f=e(PV)f-Ve.Pft (~P,U)f~~(PU)f-Uo.P$

Applying also (33), we get

By Theorem 7, P f is invariant under ( U V )f .

41. Solution of P f = O invariant under Uf and V f when P, U,
V are linearly independent. We employ the notations (3) and (5).
Then A is not identically zero since P, U, V are linearly independent (5 32).
Hence we can solve relations (3) for the three partial derivatives as linear
functions of P, U, V. Substituting the resulting values in the expression
for the commutator (U V )f as a linear function of those partial derivatives,
we obtain a relation of the form

in which 1, p, v are known functions of x, y, z. Since P f = 0 is in-

variant under U f and V f , we have (34). Insert the expression (35) for
( U V ) and apply (31) and (33). We get
 -
338 L. E. DICKSON.

This must be identical with z P f by (34). But P, U, V are linearly in-

dependent. Hence PI. r 0, P p 0, so that I and p are solutions of
Pf = 0, which is invariant under Uf, Vf. Thus, by (27),

(36) I, r , UL, U P , VI, VP

are all solutions of Pf = 0. If any two of these six functions are in-
dependent, we have lound the complete solution of Pf = 0 by algebraic
processes and differentiations.
There remain two cases. Either I. and p are both constants, a case
treated under (ii); or else I. (for example) is not a constant and the remaining
five fi~nctions(36) are all functions of I. The latter case will be treated &st.
(i) Let I. be a solution, not a constant, of Pf= 0, such that 271 = g(I),
V I = h (i.). Then the infinitesimal transformation *

leaves Pf = 0 invariant. For, by (31),

(12 Li, P )f E h ( U P )f -Ph . Uj;

and P h r Pg = 0, since PI. - (gT7,P )f

0. Hence, by (33),
g ( V P )f -P.9. Vj;

Furthermore, TVI = h g - g h -
0. Hence we may apply 5 38 (with W
in place of U) and !ind by quadratures a common solution, in addition to I,
of the complete system Pf = 0, Wf = 0.
For example, the equation

is invariant under the two infinitesimal transformations

since ( U P )f = 0, ( V P )f = 0. Also, P f , Uf, and f are linearly independent since

i7
the determinant of their coefficients is

* If TVf = 0, then h (R) 0, g(2) 0, since U f and Vf are linearly independent

by hypothesis. Then 2 is a common solution of P f = 0, U f = 0, Vf = 0, contrary
to Theorem 1.
 DIFFERENTIAL EQUATIONS FROM THE GROUP STANDPOINT. 339
+ +
We find that (UV)f = k Vf, k = 2/(y 22 1). Hence k, and also R = y 2.2, is +
a solution of Pf = 0. Also, UR = 2, VR = 0. Hence t,he problem falls under the
present case (i) with W f = - 2 Vf. We remove a factor and employ

To proceed as in 5 38 (with W in place of U), we introduce the new variables x, y, @ = y + 2 z,

and find that Pf and W f become

is an exact differential. In the integration we regard @ as a constant, write t for

x -$ y -$ 0 ,and eliminate x. We get

Returning to the initial variables, we obtain the second solution

(ii) Let 1 and p in (35) be constants. Without loss of generality we

may take p = 0. For, if p 0, P f = 0 is invariant under

A
U1f=-UfSVf, V,f=-Uf,
P
for which, by (35),
(UlVlIf = - - ( V U ) f = ( U n f =P jyA ~ f + + f ) + v p f ,
= r U l f + l,Pf,
and this is of the form (35) with p = 0. Hence we may write

(37) (UV)f = cUf + ~ ( 2y ,

,z)Pf (c a constant).
Since P f = 0 is invariant under U f and Vf, we have

(38) ( U P )f =e P f , (VP)f upf.

The f i s t relation shows that P f = 0 and U f = 0 form a complete system;
let QJ be their unknown common solution. The second relation and (37)
may be written in the forms
340

-
Take f =. cD and apply PO - - -
L. E. DICKSON.

0, UCD 0. We see that P(V@) -

U ( T @ ) 0, so that 1'0 is a solution of our complete system Pf = 0,
IiTJ' = 0, and hence is a function of CD alone, T'CD
0,

Y(CD). If @(@) were

identicallr zero, Vf = 0, Pf = 0, and Uf = 0 would have a common
solution CD, not a constant, and hence by Theorem 1 would be linearly
dependent, contrary to hypothesis. Hence +(O) 0. As in 8 15, we can
therefore determine a function x(@) such that V x r 1. Hence there exists
a function 31 for whicli

Employing the expressions (3)for Pf, Uf, T'f, and recalling that the
determinant A is not identically zero, we see that (39) can be solved as
linear equations for

and their values substituted in

But the desired expression for d x may be found more simply by a device.
Since

are homogeneous in (40),the determinant of their coefficients must be

identically zero :

Employing the determinant A in (5), we get at once

 DIFFERENTIAL EQUATIONS FROM THE GROUP STANDPOINT. 341
Hence

I f c = 0 in (37), we see from it and (38) that Pf = 0 and Vf = 0

form a complete system having a common solution I+ for which P(UI+) 0 ,
V(Uv) 0 , whence Uq is a function of I+ alone. Without loss of
generality (5 15), we may take Uv = -1. As before

Ids tly dzj

Since U Xr 0, @ is independent of x.
If c #= 0 , it remains to find a solution of Pf = 0 independent of the
solution X. Since is a common solution of Pf = 0 and Uf = 0 , we
may proceed as in 5 38 and find by quadrature the desired new solution.
THEOREM.If Pf = 0 is invariant under two given injnitesimal trans-
formations Uf and Vf, such that the latter and Pf are linearly independent,
the integration o f Pf = 0 requires only qtcadratures.
As an example under case (ii), the equation

(41)
is invariant under
Pf = 2 2 -aa$xf2 - - ( x a+
a f2 z ) -
y
af
az - 0

since ( U P )f = 0, ( V P )f = 0. Here ( U V )f =0 and

so that our problem fails under case (ii). Since

a common solution of P f = 0 and Uf = 0 is

=J ax-2d.y
Py-x
= - log (x -2 y).
342 L. E. DICKSON.

While we might compute the integral 9, it is simpler to proceed as suggested nnder

+
the case c 0 and employ the new variables x, w = x-2y, z. Then Pf = 0 becomes

which is invariant nnder Uf. From the resulting integrating factor, we conchde that

i +
is an exact differential; in fact, of log (x2 4x2). Hence x - 2 y and x2 4x2 are +
two independent solutions of Pf = 0.
Further examples of greater intrinsic interest are solved by the method of this case (ii)
in 5 49. Two instmctive, but louger problems of geometrical origin are treated by this
method by Lie-Scheffers, Differentialgleichungen, pp. 453-456.
42. Solution of Pf = 0 invariant under Uf and Vf when P,
U,V are linearly dependent. It is understood (end of § 38) that Uf
is not the product of Pf by a function of x,y, z. But Pf, Uf, Vf are
by hypothesis connected by a linear relation. The coefficient of Vf is
not identically zero and may be divided out, giving

Since Pf = 0 is invariant under Uf and Vf,

(UP)f-ePf, (VP)fEuPf.
Hence by (31),

Thus PcD 0, since otherwise Uf would be a multiple of Pf. Hence

we know in advance the solution Q, of Pf = 0. We may exclude the
case in which Q, is a constant, since Vf in (42) is then not regarded as
essentially distinct from Uf with respect to Pf = 0. In fact, when
Pf = 0 is invariant under Uf, it is necessarily invariant under Q, Uf vPf, +
for every constant Q, and function v, since (as just shown)

and no new knowledge is added by the assumption that Pf = 0 is in-

variant also under (42).
 -

DIFFERENTIAL EQUATIONS FROM THE GROUP STANDPOINT. 343

We seek a solution of P f = 0 which shall be independent of the known
solution 0. Since ( U P )f e Pf, it is furnished by U @ if U 0 is in-
dependent of 0. Henceforth, let U @ be a function of @.
For definiteness, let the variable z actually occur in 0, and introduce
x, y, 0 as new variables. Since P@= 0, P f and U f become

If U @ = 0, Ulf gives us an integrating factor of the ordinary differential

equation corresponding to Pl f = 0. But if U O $ 0, Ul f alters 0, which
occurs only as a parameter in PIf = 0 and is evidently* of no assistance
in the integration of PIf = 0. We must therefore solve the corresponding
ordinary differential equation in x and y without any effective assistance
from the known infinitesimal transformations.
THEOREM.I f Pf = 0 is invariant under two essentially distinct iqfinit-
esimal transformations U f and Vf such that the latter and Pf are linearly
dependent, one solution @ is known in advance from (42) and the determi-
nation of a second sok~tion requires, ullten U@ is a function of @, the
integration of a n ordinary di$erential eqz6ation of tlie j r s t order in two
variables.
To give a brief, but highly artificial, illustration, note that the equation (41) is invariant
also under
Wf
x+42
= Z Y + X Z - ~ ~ aza-af

+
Since U f = @ TVf, where @ = x y z z - y2, @ is a solution of Pf = 0. Then
+
U @ = x 2 4 x 2 = 4 is a new solution. From them we obtain 4 - 4 @ = (x - 2 Y ) ~ ,
and hence have our former solutions.

CHAPTERIV.

Ordinary differential equations of the second order.

The primary object of this chapter is to show how to utilize the know-
ledge that a differential equation of the second order is invariant under
one or two known infinitesimal transformations in order to simplify the
work of its integration. In the f i s t case it is necessary to solve auxiliary
differential equations of the first order, but in the second case we succeed
by quadratures alone. The methods may be readily extended to differ-
ential equations of higher orders.
+
* For example, A (x,y) a f / ax B (x,y) a f /a y = o is invariant under a f / a z what-
ever be A and B. Since the invariance implies no information concerning -4 and B, it
cannot aid in solving the equation.
3.14 L. E. DICKSON.

The important secondary objects of the chapter are the development of

a simple theory of difl'erential invariants, an introduction to the theory of
closed system of infinitesimal transformations, including the four canonical
types of the systems determined by two infinitesimal transformations, and
the determination of the number of linearly indspendent infinitesimal trans-
formations vhich leave a differential equation invariant.
43. Second extension of an infinitesimal transformation Uf.
We may best utilize the method employed in 5 23 to find the f i s t exten-
sion U'f if we do not employ the explicit fractional expression for yi, but
use the general notation

(1) XI = @ (x, y, a), Y; = x (x, Y, y', a).

Then y" = dy'ldx is transformed into

Denote the final fraction by NID. We desire the value of its derivative
with respect to a for a = a,. For a = a,, (1) reduces to the identity
transformation rcl = x, yi = y'. Hence

QJx= 1, 0,= 0, xx= 0, = 0, xy1= 1, N = yRl D = I, when a = h.

Since dxlcla reduces to q' and d o I d a to 5 by 5 23, we have

Hence

The second extension of Uf is therefore

where
d7 dS
-
(3) ,/f = --y'
dx d x = qx+ ( q ~ - E z ) y ' ~ - ~ ~ yl ' ~
 DIFFERENTIAL EQUATIONS FROM THE GROUP STANDPOINT. 345

By an argument analogous to that in $ 28, we obtain the

THEOREM.The differential eqzbation F(x, y, y', y") = 0 of the second
order is invariant ztnder the injinitesimal transformntion Uf if and only
if the ordinary eqztation F = 0 in .four i~zdqendentvariables x, y, y', y"
is invariant tmder the second extension U1'f of Uf, and Jzence if U"F
vanishes for all sets of vnlztes of the foztr variables satisfying F = 0.
For example, the first extension of

was found at the end of $ 26. Its second extension is

Since U y " = - 3 n y ' y " is zero when y" = 0, the differential equation y" = 0 is
invariant under Cf. This is geometrically evident when n = - 1, since Uf is then
the i n h i t e s i ~ a lrotatiou about the origin, while the system of all straight lines in the
plane (the integral curves of y" = 0) is invariant under all rotations.
4. Differential invariants of the second order of UJ: By 5 26,
there exists an ordinary invariant z~(x,y) of Uf, i. e., a solution of Uf = 0,
and a differential invariant v(x, y, y') of the first order, i. e., a solution
involving y' of U'f = 0, where G'f is the first extension of i7$ But it
is not necessary to proceed by a similar process of integration in order to
obtain a differential invariant w of the second order of rA i. e., a solution
involving y" of U f = 0, where U1'f is the second extension of lif. In
fact, Lie gave the following device to find w by differentiation.
If a and b are any constants,

is a differential equation invariant under Uf since its left member is a

differential invariant. Keeping a fixed, but making b vary, we obtain an
infinitude of differential equations, each invariant under UJ Thus each
has a family of integral curves which are permuted by Uf. The totality
of the curves of the inhitude of families is therefore invariant under Uf.
This totality of curves is the set of integral curves of a differential equation
of the second order which is invariant under Uf and is obtained by diffe-
rentiating (5), since we obtain a result lacking b and hence true for all
the curves in question. We get

dv-adu =0
346 L. E. DICKSON.

or w-a = 0, if we employ the abbreviation

Since w-a = 0 is invariant under 'Uf, we conclude from § 43 that

vanishes for all sets of solutions of w -a = 0. But U"w does not involve
the arbitrary constant a which appears in to- a = 0. Hence the preceding
condition requires that U"u:be identically zero. *
Hence w is invariant under U1'$ Moreover, the coefficient of y" in (6)
is not identically zero since y' occurs in v by hypothesis. Hence w is
a differential invariant of the second order of U f .
THEOREM. Given an ordinary invariant u(x,y) and a diferentinl invariant
v (x,y, y') of the first order of an infinitesimal transformation Uf, toe obtain
by diferentiation a di$erentiat invariant w = d v l d u of the second order of U f .
SimiIarly, differentia1 invariants of higher orders are furnished by further
differentiations :
d2v d3v
"'

45. Integration of all differential equations of the second order

invariant under Uf. By an argument like that in § 30, we see from
the last theorem that the most general differential equation of the second
order invariant under Uf is obtained by equating to zero a function of
u, v, w and hence its solved form is

dv -
- - F ( u , v).
du
* While this is rather evident, a formal proof is readily supplied. The expression (6)
for w shows that it is a linear function ky"+ I of y". In the symbol (2) for U"f, 7" is
given by (4j and is linear in y " , so that U"w is a linear function c y W + d of y". By means
of w - a = 0, we eliminate y" from U w = 0 and obtain

This must hold for every a (as weil as every x , y, y'). Hence c = 0 and then d =0. In
other words, E w is identically zero.
 DIFFERENTIAL EQUATIONS FROM THE GROUP STANDPOINT. 347

Suppose we have integrated this differential equation of the &st order

in the variables ti, v and found an integral 0 ( w , v) = c. The latter is
a differential equation of the first order in x,y which is invariant under Uf
(since 24 and v are invariant), whence an integrating factor is given by 5 15.
EXAMPLE1. Let l7f be y a f1 a y, so that e = 0, 7 = y. Then 7' = y',

Evident solutions of U'f = 0 are u = x, v = y'ly. Then

Hence every differential equation of the second order invariant under y a f / a y is of the
form

An interesting case is that in which 9 is a linear £unction Q (x) + N ( z )y'l y of y'l y.

Then we obtain the homogeneous linear Werential equation

The first step in our method of integration consists in reducing (8) to the type (7) by
the substitution x = u , y'l y = v, which imply

by the above relations. Thus (8) becomes a Riccati equation

Let vo(u) be a particular solution und write v = llz + vo, where z= z (u) is the new
dependent variable. Then

which is a linear differential equation solved in § 16.

Exax~m2. Any linear differential equation of the second order

can be integrated by quadraturea when there is known a particular integral z(x) of the
+
corresponding abridged equation (8). For, if y is any integral of (9), y c z is evidently
345 L. E. DICKSON.

an integral when c is any constant. In other words, (9) is invariant under the group of
+
transformations x,= x, y, = y c z , which is generated by the infinitesimal transformation

In order to apply our general theory, we need in addition to the evident invariant
u = x of Uf also a differential invariant v of the first order, i. e., a solution of

a f + 2' (x)-
L'. f = z ( x ) - a f = 0.
a9 ay'

Hence v is an integral of the corresponding ordinary differential equation

W e may therefore take v = z y' -z1y. Then

is a differential invariant of Vf of the second order. By the general theory we know

that our equation (9) can be given the form (7), viz.,

To obtain the latter we recall that z is an integral of (8):

Hence we multiply the latter by -y, and (9) by z, and add. W e get

By 5 16, an integral of this linear differential equation of the first order is

Denote the right member by @ (x). Then

 D I F F E R E N T I A L EQUATIONS FROM T H E GROUP STANDPOINT. 349
By the method used in these examples we obtain the

TABLE
OF DIFFERENTIAL EQUATIONS OF THE SECOND ORDER INVARIANT
UNDER THE ACCOMPANYING IBFINITESIMAL TRANSFORMATION.

the last three being obtained from the preceding three by interchanging
x and y and hence replacing y' by lly', y" by -y"l~j'~.
46. Second method of integration. Let the equation be

(10) Y" -01(x)y, y') = 0,

in solved form. Since
dx -
- _ -dY'-y -
dy'
1 W(X, Y , Y') '
the corresponding partial differential equation is
350 L. E. DICKSON.

The first and second extensions of U f are

only if U " F -
By the theorem of 5 43, F = yl'-o = 0 is invariant under U f if and
g'' - U' w vanishes identically in x, y, y' after yl' has
been replaced by w. Inserting the value (4) of q", we see that t,he con-
dition becomes

The sum of the first three terms is Pq'. Hence (10)is invariant under U f
if and only if

(13) Pql-w-- dS U'w ZE 0)

dx

identically in z, y, y'. The commutator of U ' f with PJ'is

Replacing q1 by its value (3),we get

if and only if (13) holds. Hence by Theorem 7 of 5 37 we deduce the

THEOREM. The diferentiaZ equation yN = o (x,y, y') is invariant under
the irfinitesimal transfmmation U f i f and only i f the carresponding partial
di$erentiaZ qtcatim (1 1) is invariant under the f i s t extension U ' f of U$
A convenient form of this condition is (12).
+
EXAMPLE.The system of conics a x 2 b y2 = 1 is evidently invariant under tbe trans-
formations x, = r x , y, = s y. Take e aa a function of r and differentiate the two
equations with respect to r. Write n for the value of deldr when r = 1. We obtain
the infinitesimal transformation
 DIFFERENTIAL EQUATIONS FROM THE GROUP STANDPOINT. 361

To obtain the differential equation of the conics, differentiate a x 2 + by2 = 1 twice and
eliminate a and b. We get
x2 y2 -1
5 YY' 0 = 0,
1 ~'~+yy'' 0
or

The corresponding partial ditrerential equation (11) is

To check that it is invariant under U'f, note that

(U'P) f = - Pf.
Thus Pf = 0 and U'f = 0 form a complete system and this fact simplifies the solntion
of Pf = 0. The simplest U'f is that with n = 1:

U'f = U f = x - + y -a.f af
ax ay
Evident solutions of U'f = 0 are u = y l x , v = y'. Some function P ( u , v) of these
must be a solution also of Pf = 0. Thus

Removing the common factor y'- yiz, and multiplying by y, we get

which has the evident solution uv = yy'lx.

To find another solution of Pf = 0, we employ the variables x, y, z = yyf/x. The11
Pf = 0 and U'f become

Since the former is invariant under the latter, x z d x - y d y = 0 has the integrating
factor 1 l ( d z -y') by 3 15, whence x2z - y2 is a solution. From the resulting two first
int-
3% = C, X ~ ~ ' - ~ Z = d
x

of (15), we eliminate y' and obtain the general integral cx2---y' = d of (15), and hence
the initial conics with altered parameters.
352 L. E. DICKSON.

47. Linearly independent or d e p e n d e n t infinitesimal trans-

.
formations. We shall call U,f, .. , U,f linearly dependent if there exist
.
constants k,, . . , k, not all zero such that k, Ul f +. +
. kUTf r 0, but
linearly independent if no such constants ki exist. In the first case one
of the Uif can be expressed linearly in terms of the remaining U's; if
the latter are linearly dependent we repeat the process. Hence we may
assume that U,f, . . . , Gkf, for example, are linearly independent, while
.
Uk+lf, . . , LT8f are expressible linearly in terms of the former.
LEMMA.If yr' = o (2, y, y') is invariant under Ulf,. . . , Ukf, it is
invam'ant ztnder every transfmation

(16) Uf r fi U,f +..-+ c,LTkf (cl, . . . , ck constants),

which is Zineag-Zy dependent on them.
.
For, by 5 46, their f i s t extensions U; f, .. . Uif leave (11) invariant,
so that
(17) ( U ; P ) f _ e 1 P f , .... (ULP)f=ekPf.
We see at once from (3) that the first extension of (1G) is

(U'P) f r cl(UiP)f + + ck(ULP)f (c1e1-I- .. + ckek)Pf-

Hence by Theorem 7 of 5 37, U'f leaves Pf = 0 invariant. By 5 46,
Uf therefore leaves y" = w invariant.
Hence (16) may be discarded since it furnishes no information about
y" = w which is not already implied by its invariance under U,f, . . ., Ukj:
48. M a x i m u m n u m b e r of linearly independent infinitesimal
t r a n s f o r m a t i o n s Uf leaving yN= w invariant. We shall assume that
the given function w (2, y, 9') may be expanded into a series involving only
positive powers of y'. If we insert the expression (3) for q' into (12), we get

in which subscripts denote partial differentiation. Replace w by its ex-

pansion in a scries. Then the total coefficient of each power of yr in our
identity. must be zero. By the terms free of y' and the coefficients of
yr, yl', yf8, we evidently get
 DIFFERENTIAL EQUATIONS FROM THE GROUP STANDPOINT. 353

where each fi is a linear homogeneous function of

whose coefficients are known functions of x and y. By the coefficients

of y'" y f 6 , . . ., we obtain linear relations involving the functions (20) which
together with (19) are necessary and sufficient conditions on E and q in
order that

shall leave y" = o invariant.

The eight relations obtained by differentiating each of the four equations (19)
partially with respect to z and y in turn are seen a t once to give the
values cf the eight partial derivatives T ~ I qm,
, . . ., Ew of the third order
expressed as linear homogeneous functions of (20) and the six second partial
derivatives of 5 and q.
If we assign the values a t x = 0, y = 0 of the functions (20) and Em
and Em, we may compute by (19) and their successive derivatives the values
of all second and higher derivatives of 5 and q, and hence determine 5 and q
by means of
f ( ~Y ,) = f ( 0 , 01 + ~ f ~ + ~ f ~ + a ~ ~ f ~ ...,
+ ~ ~ f ~ + 3
where fx denotes the value of fx at x = 0, y = 0, etc.
THEOREM. * A di$erentiaZ equation of the second ordw is invariant tcnder
at most 8 linearly independent infinitesimal transformations.
In the above discussion we attended only to the conditions (19) and
ignored the further linear relations between the functions (20). The number
of arbitrary constants in the general solution E, q of all these relations
may be fewer than 8. Moreover, these relations may not be consistent.
In fact, let
(x,Y?y') --
S ( X ?Y ) @'. +
Then the coe£iicient of ev' in (18) is

(qy-25x-38~yr)-[qx+(qy-tz)y1-8yy'2] = 0,
whence

--
Ey = 0, ?y -- fz
-
= -72.
*For a geometrical proof, see Lie-Scheffers, Continuierliche Gruppen, pp. 294-8. It is
proved also that nn crdinary differential equation of order r ( r > 2 ) is invariwt under
+
at most r 4 linearly independent infinitesimal transformations. This maximum is reached
for fl)= 0.
354
- + +
-
L. E. DICKSON.

Thus Zxx r 0, ;i ax b, q = a ( y -x) c, where a, b, c, are constants.

Thus (18) becomes -Fg,-qq,-;i33g 0. Take g xy. Then
n = b = c = 0, whereas Z and q are not both zero identically. Hence
-
y" = 2 y $ ev' is i~zl.ariant rmder no injinitesimal transformation.
But for 01 = 0, (18) holds if and only if conditions (19) hold with each
j , = 0. All eight partial deriratives of the third order of F and rj are
now identically zero. Hence 2 and q are polynomials of the second degree
in rr: and y. Then (19) with each f, r 0 are satisfied if and only if

Hence* y" = 0 is invariant under ezactly 8 linearly independent in.nitesinzal

frcinsjil?.n~
ations.
49. Differential e q u a t i o n of the second order i n v a r i a n t under
t w o infinitesimal t r a n s f o r m a t i o n s . We shall begin with illustrative
examples whose discussion discloses the advantages to be gained from
a knowledge of two infinit,esimal transformations each leaving invariant
the differential equation.
EXAMPLE1. W e shall treat equation (15) from a new standpoint. From the start,
we knew that i t is invariant under the infinitude of infinitesimal transformations (14), or,
what is equivalent, under the two linearly independent transformations

In other words, Pf = 0 is invariant under their first extensions

* This follows also from the fact that the integral curves of y" = 0 are all the
straight lines in tihe plane. These lines are merely permuted by the socalled projective
transformations
21 =
alx+B,y+cl ~ , x + ~ z Ycs+
y1 =
aa~:+b8y+cZ ' nax+ bay+ca

We obtain a one-parameter group by taking a,, . . ., c8 to be functions of t such that

al = b2 = cs = 1, b, = c, = az = c, = a* = b3 = 0 when t = 0 (whence zl= z, y, = y).
Then

if A,, BI, . . . denote the values of the derivatives of a,, b,, . .. when t = 0. Writing
Dl for A, - Ca, and D 2for BI - Ca, we obtain (21).
 DIFFERENTIAL EQUATIONS FROM THE GROUP STANDPOINT. 355
Hence we may apply the method of case (ii) in 5 41 with ( U V )f = 0. We shall omit
the details of the integrations since full details will be given in the next two examples,
which are more interesting than this one. We find that

whose difference is log (y2- xyy'). We therefore obtain the same results as in our earlier
treatment of this example.
+
EXAXPLE2. To integrate y" = S ( x ) y' Q (x) y R (x). +
Let z(x) and ~ ( x be) two linearly independent particular solutions of the abridged
+
equation y" = ATy' Q y (see Ex. 3). Then

is invariant under the following two linearly independent infinitesimal transformations

(8 45, Ex. 2) :

The problem f& under case (ii) in 5 41 since

For, if A = 0, then dw/w = dz/z, w = CZ, where c is a constant, contrary to tlie

assumed linear independence of z and w. Next,

Z" = Nz' + Q z, to1' = N W' +Q 10

give

Inserting these values in the formula for the common solution

(22) x =
1 " NY~+;Y+R~

A
of Pf = O a d Uf = O , we get

Since the &st integrand is'an exact Werentid, we get

 Ry interchanging z with tc (and z' with w ' ) , we deduce the second solution

Eliminating y' between the first integrals 2: = a and Q = b, me get the following general
integral of the proposed differential equation:

EXAMPLE3. Given a particular solution z ( x ) , to integrate

-4s in Ex. 2, the corresponding partial differential equation Pf = 0 is invariant under

Uf. By Ex. 1 of 5 45, i t is invariant under also

Here 3 r zy' -z' y is not identically zero in the independent variables x , y, y', since
z 0. Also ( U V )f -- Cf. Hence the problem falls under case (ii) of 5 41. W e have
(22) with R = 0. Replacing Q z by its value z" - N z ' , we find that the integrand in
(22) is the total derivative of X :

Introduce the variables I, g, v . Then Pf E 0 becomes

When v is regarded as a constant, Pf = 0 is invariant under

since the commutator is identically zero. Heuce l/z is an integrating factor of the corre-
sponding ordinary differential equation
1 S.d
- jzty +ve
2)
dx-dy = 0,
whose integral is therefore

ma, with v taken as a constant, is the general integral.

In each of these three examples, the differential equation of the second
order is invariant under two hllnitesimal transformations Uf and Vf such that

where a and b are constants, one or both of which may be zero.

 DIFFERENTIAL EQUATIONS FROM THE GROUP STANDPOINT. 357
Our next aim is to prove that, when the two given infinitesimal trans-
formations leaving invariant a differential equation of the second order do
not have the property (23), we can readily determine from them two
new iniinitesimal transformations which have that property and leave the
equation invariant. We shall then be in a position to specify the four
subcases necessary in an exhaustive treatment of differential equations of
the second order invariant under two given infinitesimal transformations.
50. First extension of the commutator ( U V ) f . lFTe have

If g is an arbitrary function of z and y, write

Hence q' = B I] - y 'B 5. We have

where the expression for e will not be needed. Hence

Evidently (CU')J' = oCf, where the expression for a will not be needed.
Similarly, (BV'jf and (CV')f are linear combinations of B f and Ctf.
By Jacobi's identity in 5 39,

The W d term of this is equal to

by (31) of 5 38, and hence is a linear combination of Bf and Cf. T h e

same is true of the second term of Jacobi's identity, which therefore gives

(24) ( B ( C ' P ) ) f ~ 1 B f i paa C

fz f ~ ~ a-f- + ~a-f y ' - f ~ ~ ~

aY
Let
358 L. E. DICKSON.

In view of the foregoing expression for q', we have

Since the two terms of the commutator ( U V )f are (5 21) the f i s t two of
the three terms of the commutator of U'f with V 1 f ,we have

in which 5 may be found indirectly as follows. We find that

in which the expression for z will not be needed. Comparing this with (24),
we see that
I=BE1, Ry1--B71-C1 5GBql-y1BF1,

so that (25) and (26) are identical.

THEOREM. T h commutator (U' V ' )f of the first atensions of two injni-
tesimal transformations U f and V f is identical with t1zJtrst atmsion (UV)'f
of their commutator ( L T Vf).
Corollary. I f y" = o (x, ?/,y') .is invariant under both U f and V f it
is invariant abo under their commutator ( U V )f.
For, the corresponding partial differential eqaation P f = 0, given by ( l l ) ,
is then invariant under their first extensions U'f and V ' f and hence (5 40)
under (U'TI)$ Since P f = 0 is therefore invariant under (UV)'f, y" = o
is invariant under ( U V )f by 5 46.
51. Closed s y s t e m of infinitesimal transformations leaving y" - w
invariant. Given several infinitesimal transformations leaving y" = w
invariant, we may discard (5 47) those which are linearly dependent on the
remaining. Hence it suffices to consider linearly independent infinitesimal
.
transformations U l f , . ., Ukf leaving y" = w invariant.
By the preceding corollary, it is invariant under (Ul U U B )whichf, we
discard if it is linearly dependent on U,f, . . ., Ukf . But in the contrary
.
case we annex it to them and obtain a larger set Ulf , . ., U k + ~ fof
linearly independent infinitesimal transformations leaving y" = w invariant.
If the commutator of any two of the latter set is linearly independent of
these k+ 1 transformations, we annex it and obtain a still larger set.
Proceeding in this manner, we ultimately reach a finite number (5 48) of
 DIFFERENTIAL EQUATIONS FROM THE GROUP STANDPOINT. 359
linearly independent infinitesimal transformations C1f, . . ., LTrf leaving
yl' = w invariant and such that the commutator of any two of them is
a linear combination of them with constant coefficients. Then all their linear
+
combinations cl Ul f . . . $ C, U , f with constant coefficients cl, ,, C , are ..
said to form a closed system" of infinitesimal transforniations.
THEOREM. I f two or more inJinitesima1 trctnsformntions leave i?zva?.ia?~t
a d i f i e n t i a l equation of the second order, they nre contained in a closed
system leaving i t invariant. In partiadar, the totality of infinitesimal t~.ans-
formations leaving i t invariant forms a closed system.
For example, yN = 0 is invariant under

since
U'f 8.f
x-+- af J7lf af
y--y12- af P f E -a+f &-., a t '
f
ay ayti ax 3.v" ax aM'
(UIP)f=O, (VIP'f=-y'Pf.
By the corollary in 50, y" = 0 is invariant also under TVf, where

This follows also from ( W ' P )f .=- P f. Here Lrf. V f , W f are evidently li~iearly
independent; their linear combinations with coilstant coefficients form a closed system since
(UV)f = Wf, (Un7)f = -2Uf, (VJV)f = 2Vf.
Note also the five closed systems formed by the linear combinations of
uf,w f ; v f , m 7 f ; u l f , PY; w l f , ~ f u;l f , rn1-f,p f .
All infinitesimal transformations which leave y" = 0 invariant are determined by (21).
Hence they form a closed system. It is not very laborious to verify this fact by showinq
that the commutator of any two of them is one of them.
52. Closed- systems determined by two infinitesimal trans-
formations. We are now in a position to prove the following statement
made a t the end of 8 49:
THEOREM 1. I f u7e know two infinitesimal transformations 1.7 and Tyf'
which leave invariant a given diferettial equation yl' = (I), foe can Jind
another inJinitesimal transformation V f leaving it invariant such that the

system, viz.,
(27)
* Often
(UV)f uuf+bt-J'-
linear combinations o f Uf and V f with constant coefficients f o r m n closed

(a, b constan fs).

called an r-parameter group of infinitesimal transformations. But it is neither

a group (in the technical sense), nor does i t have r essential paranieters, since the r- 1
ratios c,: q: .: G give the only essential parameters. Rut the closed system generates
an r-parameter group composed of cur transformations only car-' of which are infinitesimal
and constitute our closed system.
360 L. E. DICKSON.

By the proof of the Theorem in 5 51, we can find r linearly independent

inilnitesimal transformations Li,f = CTf, U2j; .... LT7f leaving y" = o
invariant such that
( U i C$ )::)J -- 2 Cijk b T k f (i, j = 1, .... r),

-
k=l

where the cijk are constants. We seek constants a, b, e,, .... e, such
that (27) shall hold for U Ul and

We have
r r
--
7

(CTV)f G C e j ( U 1 @)f 2 e j z c l j k Ukf.

~ = 2 j=2 k=1

We desire that this shall be identical with the right member of (27), viz.,

In view of the linear independence of GT1i,f, .... U,f, the conditions are

The first condition will serve to determine a after the e j are found. The
remaining r-1 conditions (28) may be written in the form

Such a system of r - 1 linear homogeneous equations in r - 1 unknowns

e, e4, ..., e, have solutions, not all zero, if and only if the determinant of
their coefficients is zero*:

* Cf. Dickson, First Conme in the Theory of Eqaatione, 19B, p. 119.

 DIFFERENTIAL EQUATIONS FROM THE GROUP STANDPOINT. 361

The expanded form of this relation is an algebraic equation of degree

r -1 in b in which the coefficient of by-I is (- I)'-'. Hence whatever
be the values of the given cijk, we have an equation actually involving b
and therefore having at least one root b. This proves Theorem 1.
The proof leads also to a more abstract result:
THEOREM 2. Any in.nitesima1 transformation Uf of any closed system lies
i n a closed system formed by the linear combinations of U f and another
transformation of the jrst system.
For examples illustrating this theorem see the end of 5 51.
We shall now obtain canonical forms for closed systems determined by
two infinitesimal transformations Rf, S f on x, y. By hypothesis, we have
+
(RS)f r I R f @ S f ,where I and y are constants. If 1= y = 0, take
U = R , V = S , whence ( U V ) f -0. If I # 0 , y = O , take U = R ,
V = I-I 8, whence (UV)f = Uf. If p # 0, take

whence (UV) f

(29)
- U' Hence in every case

(UV) f r kUf, k = 0 or 1.
By 5 22, this implies that the partial differential equation U f = 0 is
invariant under the infinitesimal transformation Vf. Hence a solution u (x, y)
of Uf = 0 can be found by quadratures (5 15). After interchanging x and y
if necessary, we may assume that x actually occurs in u(z, y). In terms
of the new variables u and y, U f becomes

where QJ is the function obtained by expressing Uy in terms of u and y.

Finally, we introduce the new variables u and

in the integration of which u plays the rble of a constant. Since Ul Y = 1,

U,f becomes a f 1a Y. Hence we can determine new variables by quadratures
such that LTf becomes*

* This choice (30) instead of 8f l a x simplifies the work in 53.

362 L. E. DICKSON.

Then, by (29),

whence

First. let ' ( x ) E 0. If li = 0, then e ( x ) is not identically zero, since

1,f' is not identically zero. Replacing e ( x ) by a new variable x, we have

But if k = 1, we replace y -I- e ( x ) by a anew variable y and get

(11) Uf = -
8.f
ay'
Vf = y-,aafy
(UV).f =- Uf.

Second, let 6 ( x ) be not identically zero. Introduce the variables

where is not a constant, Then

~ ( 2 : ) Uf and Trf become

in which theoretically x should be replaced by its value in terms of xi

obtained by solving X I = V ( x ) , but practically this replacement is not
necessary here. If k = 0, we take

and, after dropping the subscripts 1, obtain

 DIFFERENTIAL EQUATIONS FROM THE GROUP STANDPOINT. 363
But if k = 1, we can choose (D and ly so that

the conditions for which are

The latter is a linear differential equation for 0 ; by 5 16, we get

Dropping the subscripts 1, we have

(IV) Uf =-
af a fy - ,
VfZEx-+ af (rnf=Uf.
aY

-
ay) ax

THEOREM 3. Given two infinitesl'mal tramformations U f and V f o?z x

and y for which ( U V )f k UJ k = 0 or 1, we can find by qundratureP
alone new variabh sztch that the infinitesl'rnal transformations take one of
the four canonical forms (I)-(IV), which are distinguished from one an-
other by the value of k and the jhct that V f is t ? product
~ of U f by a
fi~nctwn of x and y in cases ( I ) and (II), but not in c a ~ e(111)
~ and ( IT).
Instead of first reducing Of to the canonical form (30) and then reducing Vf to one
of ite four canonical forms in (I)-(IV) without altering (30), we may perform the reductions
simplfaneously and in fact often more simply. For example, let

Our proof shows that there exist new variables x,, y, for which

Hence XI = is known. Since 0 = (U, pU) f r Up Uf, we have Up- = 0. In

terms of the new variables XI = p and yl, Uf therefore becomes

* In Lie-Scheffers, Differentialgleichungen, pp. 424-426, it is stated incorrectly that the

reduction to (IV) is not accomplished hy quadratures alone, but that we must solve a
clifferential equation (equivalent to Of = 0). This oversight was caused by the ignoring
of one of the hypotheaes, viz., (UV)= Uf.
364 L. E. DICKSON.

Hence dFe desire a solution y1 of

ayl ay
uy, = ex+?- = 1,
BY
which is equivalent to the simultaneous system*

But the first of these equations is equivalent to Uf = 0, which has the known solntion p.
Eliminating x, for example, by means of p = c, taken in the solved form x = @ (y,c), we get

53. Integration of a differential equation of the second order

invariant under two infinitesimal transformations. In view of
Theorem 1 of 5 52, we may assume that the two transformations define
a closed system, and then, by (29), select transformations Uf and V f from
the system such that (U'F)f r k U.6 k = 0 or 1. By Theorem 3 of 5 52,
new variables may be found by quadratures such that Uf and TJf take
one of the four canonical forms (I)-(IV). Let the differential equation
become y" = o (x, y, yl) when written in our new variables. We apply the
condition (12) that it be invariant under our Uf = a f 1 a y, for which
5 q - 0, and get amlay = 0. Henceforth we shall suppress this
derivative without further notice.
F o r Vj' in (I), we have 5 = 0, 9 = x, = 1. Then (12) becomes
V' o 0, or a ola y' = 0. Thus

For Vf in (II),F - -0, q = y, q' y', and (12) becomes o- 'V'u r 0, or

(ii) y" = y'f (x), logy' =J&)d x + logs, y = as$fm)"ds + b.

*Since dy, = -aax
?/I
dz+-dy
ay
ay
= Uy,.--dEx '
 DIFFERENTIAL EQUATIONS FROM THE GROUP STANDPOINT. 365

F o r T7f in (ID),Z r 1, q = 0, q1 = 0, and (12) becomes V ' w 0

or a ~ / a ; c=- 0.

F o r Vf in (IT),5 --
y' = O(x+ a),

x, q
y = J O(z+a,)dr+b.
y, q' =- 0, and (12) becomes -o - V ' w 0, o r

THEOREM.
If a diferenfial equation of the seco~zdorder is invariant under
two injnitesimab transformations, it can be integrated by quadratzcres.*
For example, consider the linear differential equation

9" = N ( x ) y y ' + Q ( 4 y + R ( ~ ) .
By Ex. 2 of $ 49, it is invariant under

where z and w are two linearly independent particular solutions of the abridged equation
+
y" = N y l Q y . The canonical. transformations are therefore (I) written in new vari-
ables x,, y,. Evidently r, = w/z. For y, = ylz, Uf and Vf take their canonical forms (I).
It remains to express the Merentid equation in the new variables. We find that

,\ , - ~
dx,
Y I zyl-yz' , d = zt"'-tozf, -= -
ax
A
z2'
dy\ -
- A ( 2 ~ " - y z " )- (29'- y r ' ) 3' = z A' (z" d - z' A')
ds d2 - A y,, 42 Y'- d2

* Oc account of the oversight mentioned in the foot-note to Theorem 3 of $ 52, Scheffers

A t a d that in case (IV) the integration by the present method requires the solution of
a diflerential equation of the first order. Later in his book (pp. 457-472) he took the
(onnecesaary) trouble to develop another method which requires only quadratures.
366 L. E. DICKSON.

after substituting the values of X and Q obtained in 5 49 by solving

zl' = Nz' + Qz, w" = Nw' + Qw

for N and Q. Hence

Since the find fraction is a function of x only, it can be expressed aa a function P(1;)
of x, only by means of the solved form of 1; = w/z. Then

Returning to the initial variables, we get

(ZW) -W Z ~ ) ~ (+I+ aw + bz.

Consider the above abridged equation. I t is invariant under

by Ex. 3 of 5 49. These are reduced to their canonical forms 0,written in new
variables XI, yl, if we take x1 = x, y1 = yiz. Then

?/ = ZYll Y' = z y ; + z'y, , y" = z y ; + 2z'y; + z"y,,

Y"-hTy'-Qy = z ~ + ( ~ z ' - z N ) ~ ; + ( z " - N ~ ' - Q z ) ~ , ,

and the coefficient of yl is identically zero, since z is a particular solution. Hence

y = zy. = .zSr" dx + bz.

CHAPTERV.

Applications to geometry and algebraic invariants.

We presuppose no acquaintance with the subjects to which we shall

apply the theory of infinitesimal transformations and groups generated by
them; on the contrary, our discussion furnishes luminous introductions to
those subjects.
First, we shall prove that two plane curves are congruent, i. e., can
be brought into coincidence by a rigid motion of the plane, only when
then have the same (intrinsic) equation between their differential invariants
 DIFFERENTIAL EQUATIONS FROM THE GROUP STANDPOINT. 367
of the second and third orders. Kext, we shall obtain very simply a complete
set of functionally independent covariants and invariants of any binary
form, and a t the same time provide an easy introduction to the complicated
algebraic theory of invariants.
54. Differential invariants and the congruence of plane curves.
We shall first determine all differential invariants of the group G of all
rigid motions in the plane. Each such motion is the product of a translation
q = .c+ a, y, = y + b, by a rotation about the origin. Hence the
group G is generated by the three infinitesimal transformations

Differential invariants of the first or second order were defined in 5 26 aiid

5 44 to be functions of x, y, y' = d y ! d s or of x, y, y', y N which are
inwraiiant under the first or second extensions of our infinitesimal trans-
formations. Evidently Uf' and yf' are identical with their extensions.
Hence a differential invariant does riot involve x or y. The second extension
of 'CVf was seen, a t the end of 5 43, to be

Hence every differeniial invariant of CJ of the first or second order is a

solution, not involving s or y, of W 1 y= 0, and hence is an integral of

To separate the variables multiply by y'. Hence every such differential

invariant is a function of

whicli is known to be the square of the curvature k of a plane curve C

defined by y = F(3:). Let t be the angle made with the positive x-axis
by the tangent to the curve C a t the point (x,y ) on it. Under any trans-
+ +
lation ,I. = r a, y, = y b, the curve (7 becomes a curve whose tangent
at (z,,y,) makes the same angle z with the x-axis. Under a rotation
through angle 0 about the origin, C becomes a curve whose tangent at
+
the rotated point makes the angle z 0 with the z-axis. Hence d z is
unaltered by all transformations of the group G. We shall exclude the
case in which C reduces to a straight line since z is then constant along
the a w e . Hence' d I l d t is a differential invariant if I is one, so that
also d I is unaltered by G. Since z = arc tan y',

In particular, taking I to be kZ in (2)? we get the differential invariantf

In view of the equation y = F(x)of the curve C, we may express the

differential invariants (2) and (4) as functions of x. Assume that Cis not
a circle, so that the cunTatureis not a constant. Hence we can solve (2)
for c in teims of k. Inserting the value in (4), we obtain a relation of
the form

Consider a second curve C, which is congruent to C; i. e., is derivable

from C by a rigid motioil in the plane. Then since ka and (4) are differential
invariants they have the same values a t corresponding points of C and Cl.
Hence (5) holds also for C,. The converse is readily pr0ved.S Hence
tzoo plat~ecurves, neiiher of which is a straight line or a circle, are cony?mt

* We may replace d t by the element of arcds along C, if the latter is not aminimal curve.
?Taking it as I, we get by (3) a differential invariant of order 4. After *L-2 such
applications of (31, we get differential invariants of orders 2, 3, . . . , n. If we equate to
zero the nth extension of W f and suppress the two terms involving af / a x , af l a y for
the reason given above, we obtain a linear partial differential equation in y', y", ... , y"),
having therefore exactly n-1 independent solutions. Hence every diEerentinl invariant
of order jn of a plane curve is a function of

:Scheffers, Theorie der Kurven, ed. 2, 1910, 1921, pp. 86-86.

 DIFFERENTIAL EQUATIONS FBOM THE GROUP STANDPOINT. 369
if and on2y if the same equation (5) ho2ds between the diflerenbid invariants
of the second and third orders of both curves.
Equation (5) is called the intmnsic equation of the curve since it implies
all those properties of the curve which are independent of the position of
the axes of coordinates.
The analogous theory for space curves is almost as simple.* But the
criteria for the congruence of two surfaces are more comp1icated.t Lie
developed.$ a theory of equivalence of n-dimensional manifolds under a con-
tinuous group.
55. Algebraic invariants and covariants. We shall first treat in
detail the case of a quadratic form

In order to give the customary dehition of invariants and covariants of

0, we apply to @ the substitution

and obtain the form

@1= 914 + 2r,x,y1 + sly:,
in which we have employed the abbreviations

By a direct computation, or by the l a k r shorter proof,

For this reason the discriminant ra-qs of @ is called an invariant of

in* 2 of @.
We shall say that a function C E C(x, y, q, r, s) has the covan'ant
p v t y with respect to a particular substitution (6) if

* Ibid., pp. 269-287.

t Scheffers, Theorie der Bhhen, 1902, pp. 351-3. -

S Lie-Scheffem, Continnieriiche Qrnppen, 1893, pp. 747-764.

370 L. E. DICKSON.

identically in q,y,, q, r, s, after ql, rl, s1 have been replaced by their

values (i'),and x, y by their values (6). We shall call C a covariant of
i n k I of Q, if it has this covariant property for every substitution (6).
In case C lacks x and y, it is called an invariant of 0. For example,
0 itself is a covariant of cP since (8) becomes cD1 = 0.
The substitution (6) is the product of the following two:

where k =)/Zj. Hence C is a covariant if and only if it has the covariant

property with respect to every substitution (9) of determinant unity and
also with respect to every substitution (10). The latter replaces 0 by
p k2$ + 2rk2x1 y, + sk2y:. Hence, by (8), C has the covariant property
with respect to every substitution (10) if and only if

identically in xl, y,, q, r, s, k. This is true if and only if all terms of

C(x, y, q, r, s) are of the same total order ro in x and y, and of the same
total degree 6 in q, r, s, and then I = $ (2d- w).
THEOREM 1. A covariant of Q, is a function which is homogeneous in x
a.nd y and hmwgeneous i n the coeffiats of cP and whish has the covam'an t
prvperty with respect to all substitzitions of determinant unity.
We shall therefore first seek the functions having the covariant property
with respect to every substitution (6) of determinant unity. The solved
form of (6) is then

(11) xl = dx-by, yl = -cx+ay, ad-bc = 1.

These equations define a transformation (g 3). The set of all transformations

(11) is readily verified to form a three-parameter group G, whose identity
transformation is given by

We obtain a one-parameter sub-group GI by taking a, b, c, d to be

functions of a parameter t such tbat ad- b c = 1 and such that (12) hold
 DIFFERENTIAL EQUATIONS FROM THE GROUP STANDPOINT. 37 1

when t = 0. Write a, 8, y, 6 for the values when t = 0 of the derivatives

of a, b, c, d with respect to t. Differentiate ad-bc = 1, take t = 0 ,
and apply (12); we get d + a = 0 . Hence by ( l l ) ,

so that the infinitesimal transformation which generates GI is

where

The one-parameter groups generated by these infinitesimal transformations

are composed of the respective transformations

According as a # 0 or a = 0, (11) is equal to the product

where S = &-I PI&-I denotes XI = - y, yl = x. Hence a function

which has the covariant property with respect to each of the inhitesimal
transformations (13) will have that property with respect to group C.
By means of the solved form ( 6 ) of equations (11) we obtain from (D
the coefficients (7) of 0,. The transformation (11) is said to induce the
transformation (7) on the coefficients q, r, s of (D. The combined trans-
formation (11) and (7) on the five variables x, y, q, r , s will be called
a total transformation. Thus by ( 6 ) with D = 1, a function C has the
covariant property with respect to a substitution (6) of determinant unity
if and only if C is invariant under the corresponding total transformation (11)
and (7). Here the term invariant has the sense d e h e d in 8 24 and used
in the preceding chapters. Hence we seek the functions C which are
invariant under each of the following three total intinitesimal transformations
 Id.
E. DICKSON.

To compute Uf, for example, we recall that U,f obtained from (11)
by assuming that, when t = 0, the derivatives of a, b, d are zero, the
derivatiye of c is -1, and that a = d = 1, b = c = 0. Hence, by (7),

Thus the functions having the covariant property with respect to the group G
are the solutions of

The latter form a complete system since

Our complete system of three equations in five variables has exactly 5-3
or 2 independent solutions (5 34). One solution, 0 itself, was known in
advance. The discriminant r2--qs of cD is easily verified to be a solution.
These solutions have the homogeneity properties in Theorem 1, and hence
are covariants of @.
THEOREM 2. All covariants of a binary quadratic form (D are functions
of 0 and its discriminant.
This theory is readily extended to any binary form

in which binomial coefficients (;)have been preked to the literal coefficients aj.
We may avoid the labor of applying the substitution (6) to (D in order
 DIFFERENTIAL EQUATIONS FROM THE GROUP STANDPOINT. 373
to obtain the coefficients At of the new form Dl. For example, let the
infinitesimal transformation be U,f in (13), whence

Then d O l / d t is

which is identically zero since @, is equal to (D, which is independent of t.

Since the sums by colums are zero,

Taking t = 0, we obtain the first of the following three formulas:

af ~af af
(18) T7f 3 y x - vf, v
a al aaf
f ~-+2al---+...+na,,-1----;
a a,

We again have (16). The complete system U f = 0, Vf = 0, Wf = 0

.
in n+ 3 variables x,y, a,, . . , an has n independent solutions (§ 34). Hence
all functions having the covariant property with respect to the group (6)
are functions of n independent ones. Of the latter, n =.- 2 may be chosen
free of x, y (i. e., having the inveriant property), when ,n 7 2, since ILf = 0,
v f = 0, w f = 0 form a complete system in the variables uo, . , an and ..
+
hence have n 1-3 independent solutions. In fact, these three equations
are independent if n >2 (but dependent* if n- 2, which accounts for the
invariant of a quadratic form a ) . Consider a solution of these three equations
which is either a polynomial or an infinite series, and write it in the form
* The determinant of the coefficients in the last three columns of (18)is zero identically.
374 L. E. DICKSON.

+ + .
total degrees in G,. . , a,,. Since U Sr 0 and UP,,UP,,
-
S = PI Pg . .. where P,,P,, . . are homogeneous polynomials of distinct
. . . . have the same
distinct degrees as PI,P,, . . ., respectively, we have u PI 0, tt PpE 0, . ..
Hence each Pi is a solution of the system u = 0, v = 0, w = 0. Hence
all solutions are functions of n-2 solutions each of which is a homogeneous
.
.
polynomial in a, . . , a,, and hence is an invariant by Theorem 1.
M7e saw that all functions having the covariant property with respect
to the group (6) are functions of n independent ones, and have now found
that n -2 of these may be chosen as invariants when n > 2, Hence we
need two covariants actually involving x and y. As one of these two we
may take 0 itself, and as the other we may select the Hessian H of 0, viz.,

which is readily verified* to be a covariant of index 2 of 0. To show

that 0 and H are independent functions when n > 2, take the case
0 = F1 y ; then

TIBOREM 3. All covariants and invariants of a binary form of degree

n (n > 2) are functions of the form itself, its Hessian, and n -2 homogmous
polynomial invariants.
For small values of n it is easy to solve our complete system u = 0,
v = 0, w = 0 and hence obtain the invariants. We first solve vf = 0,
which is the condition that f shall have the invariant property with respect
to all the transformations Q k in (14). Such an f is called a seminvariant
of 0. The equations of Q-t may be written

This replaces rD by 0' = a: e+n aj q-l y, + ..., where

&=ao, af=tao$al, a; = t h + 2 t a l + a n ,
a; = t 8 m + 3 P a ~ + 3 t a z + a a ,
a: = t4ao+4Pa~+6t2a2+4tas+ar,
* Dickson, Algebraic Invariants, 1914, pp. 11-12.
 DIFFERENTIAL EQUATIONS FROM THE GROUP STANDPOINT. 375
the law of formation of these equations being evident. Eliminating t between
the second and third equations, we get

so that the second member is a seminvariant; it is evidently the value of

d when t is chosen so that 4 = 0, viz., t = - a ~ l a ~ . Similarly, when
this value of t is inserted in the expressions (21) for a; and a:, we obtain
serninvariants.* To avoid the denominators a,, write Ad = cci,-l a;. We get

Since the transformattion (20) with t = -allno is of determinant unity

and transforms @ into 0' having = 0, any seminvariant S of 0 has
the property

(23) S(aO,. - a. aJ = 8(no, 0, a;, . ..)

and hence is a function of ao, A2, . . . , An. This follows also from the fact
that they are independent solutions of 2.f = 0.
.
A function f of ao, A2, . ., An is a solution of wf = 0 if

We find that

F o r n = 2, (24) becomes af l a a , = 0, whence f is a function of A,.

Since As is a solution also of uf = 0, it is invariant, and every invariant
is a function of A,.
For n = 3, (24) becomes

* For a proof of this principle, see Dickson's Algebraic Invariants, p. 47.

376 L. E. DICKSON.

whose solutions are evidently the functions of T = 4 I a ; and s = A,la,.

We readily verify that

114= ( A , + 2 a, $)lao, u A, = ( 3a, A, - 6Ai)la,,

= a , us = - 6 4 1 ~ : .

Hence if a function f (r, s) is a solution of uf = 0,

Hence every invariant of a binary cubic @ is a function of

which is the discriminant of 0, being that of the reduced cubic

a,$+ 3 4 a;1z,y;+A,a;2@.

For n = 4 , the solutions of (24) are evidently the functions of

We readily verify that

Hence a solution f ( r , s, t) of 1bf = 0 must satisfy

The corresponding system of ordinary differential equations is

 DIFFERENTIAL EQUdTIONS FROM THE GROUP STANDPOINT. 377

An obvious solution is 3r2+ t = 1. Elimination of t gives

Inserting the expression for I , we get

Hence every invariant of the quartic is a function of

By (23), every invariant which is a polynomial in a,, . . ., a, is the quotient

of a polynomial P in a,, As, Ag, & by a power a; of h. The invariant
was shown to be a function f ( ~ s,, t ) of the expressions (27). Without
altering a,,, a*, n4, let us change the signs of al and as. Then A, and A,
are not altered, while A3 is changed in sign. Since r, s, t are not altered,
the expression Pla; being equal to f (r, s! t) must remain unaltered, whence P
involves only even powers of -4,. Thus PIa; is a sum of terms c a,* d d' @,
where B, y, d are integers 2 0 and u is a positive or negative integer.

evidently expressible as a polynomial in Y, t 3 1.' I, s j-r3- t r

Being a function of I and J only, the invaliant is a polynomial in I
and J.
-
Since this sum is equal to a function of r , s, C only, each a = 0. Hence
every polynomial invariant is equal to a polynomial in I., s, t, which is
+ -J.

THEOREM 4. E~e).y(polynmnial) invariant oj' a binary qztadratic m cztbic

form is a (polyr~omial)function of its discriminant. Every (po2ynomial) in-
variant of a binary quartic form. iis a (polynomial) jitnction of tIte t x o
invariants (29).
While all covariants of the binary cubic j' were proved above to be
functions of ,f; its Hessian H , and its discriminant D, it does not follow
that every polynomial covariant is a polynomial in j; H, D. In fact,
DfP-4HB is the square of a polynomial covariant G (the Jacobian or
functional determinant of f and H ) , while G itself is not a polynomial in
j; H, D. The algebraic complete system of covariants of f contains one
(and only one) additional covariant G not needed in the functionally com-
plete system f, H, D. 9 like result holds for the quartic f, where now
IjaH-.Jf'Y-4HVs the square of the Jacobian G of j' and H.
 For the binary quintic j' the algebraic coiuplete system contains 23
covasiants, four of which are invariants, the square of the oue of degree 18
being a polynomial in those of degrees 4, 8, 12. Those three, together
with f and its Hessian, form the functionally complete system in accord
with Theorem 3.
I t is customary to speak of the operwt,ors L'f and V-i' ;IS n ~ t ~ z i l ~ i l n t o t . . ~
of covariants. In the algebraic theory, no use is inade of M-j; which is
the commutator of U j and T 7 j : But we found above that its einpl~~vrne~it
materially simplifies the con~putations.
There is an interesting application of continuous g o u l ~ sto liypercompler
numbers. An elementary account of this application has been given* b-
the writer.
* 011 the relations between lineal. algebras uld col~ti~luousgroups. Bull. Amer. Natk.
Soc., vol. 23 (1915), pp. 53-61.