CO
■apf
INTRODUCTORY TREATISE
ON LIE'S THEORY
OF FINITE CONTINUOUS
TRANSFORMATION GROUPS
BY
JOHN EDWARD CAMPBELL, M.A.
FELLOW AND TUTOR OF HERTFORD COLLEGE, OXFORD
AND MATHEMATICAL LECTURER AT UNIVERSITY COLLEGE, OXFORD
OXFORD
AT THE CLARENDON PRESS
1903
HENRY FROWDE, M.A.
PUBLISHER TO THE UNIVERSITY OF OXFORD
LONDON, EDINBURGH
NEW YORK
PREFACE
TN this treatise an attempt is made to give, in as
-1- elementary a form as possible, the main outlines
of Lie's theory of Continuous Groups. I desire to
acknowledge my great indebtedness to Engel's three
standard volumes on this subject; they have been
constantly before me, and but for their aid the present
work could hardly have been undertaken. His Con-
tinuierliche Gruppen, written as it was under Lie's
own supervision, must always be referred to for the
authoritative exposition of the theory in the form in
which Lie left it. During the preparation of this
volume I have consulted the several accounts which
Scheffers has given of Lie's work in the books entitled
Differ ential-gleichung en, Continuierliche Gruppen, and
the Beruhrungs-Transformationen ; and also the inte-
resting sketch of the subject given by Klein in his
lectures on Higher Geometry. In addition to these
I have read a number of original memoirs, and would
specially refer to the writings of Schur in the Mathe-
matische Annalen and in the Leipzig er Berichte. Yet,
great as are my obligations to others, I am not with-
out hope that even those familiar with the theory of
Continuous Groups may find something new in the
form in which the theory is here presented. Within
the limits of a volume of moderate size the reader
will not expect to find an account of all parts of the
subject. Thus the theory of the possible types of
group-structure has been omitted. This branch of
iv PREFACE
group- theory has been considerably advanced by the
labours of others than Lie ; especially by W. Killing,
whoso work is explained and extended by Cartan in
his These sur la structure des groupes de transforma-
tions Jinis et continus1. A justification of the omission
of this part of the subject from an elementary treatise
may perhaps also be found in the fact that it does
not seem to have yet arrived at the completeness
which characterizes other parts of the theory.
The following statement as to the plan of the
book may be convenient. The first chapter is in-
troductory, and aims at giving a general idea of
the theory of groups. The second chapter contains
elementary illustrations of the principle of extended
point transformation. Chapters III-V establish the
fundamental theorems of group-theory. Chapters VI
and VII deal with the application of the theory to
complete systems of linear partial differential equa-
tions of the first order. Chapter VIII discusses the
invariant theories associated with groups. Chapter IX
considers the division of groups into certain great
classes. Chapter X considers when two groups are
transformable, the one into the other. Chapter XI
deals with isomorphism. Chapters XII and XIII
show how groups are to be constructed when the
structure constants are given. Chapter XIV discusses
Pfaff 's equation and the integrals of non-linear partial
differential equations of the first order. Chapter XV
considers the theory of complete systems of homo-
geneous functions. Chapters XVI-XIX explain the
theory of contact transformations. Chapter XX deals
1 See the article on Groups by Burnsicle in the Encyclopaedia Bri-
tannica.
PREFACE v
with the theory of Differential Invariants. Chapters
XXI-XXIV show how all possible types of groups can
be obtained when the number of variables does not
exceed three. Chapter XXV considers the relation
subsisting between the systems of higher complex
numbers and certain linear groups. I have added
a fairly full table of contents, a reference to which
will, I think, make the general drift of the theory
more easily grasped by the reader to whom the sub-
ject is new.
It now remains to express my gratitude to two
friends for the great services which they have ren-
dered me during their reading of the proof-sheets.
Mr. H. T. Gerrans, Fellow of Worcester College,
Oxford, at whose suggestion this work was under-
taken, found time in the midst of many pressing
engagements to aid me with very helpful criticism.
Mr. H. Hilton, Fellow of Magdalen College, Oxford,
and Mathematical Lecturer in the University College
of North Wales, has most generously devoted a great
deal of time to repeated corrections of the proofs,
and suggested many improvements of which I have
gladly availed myself. With the help thus afforded
me by these friends I have been able to remove some
obscurities of expression and to present the argument
in a clearer light, though I fear I must still ask the
indulgence of my readers in many places. Finally
I desire to thank the Delegates of the Oxford Uni-
versity Press for undertaking the publication of the
book, and the staff of the Press for the great care which
they have taken in printing it.
J. E. CAMPBELL.
Hertford College, Oxford.
September, 1903.
CONTENTS
CHAPTER I
DEFINITIONS AND SIMPLE EXAMPLES OF GROUPS
SECT. PAGE
1-3. Operations defined by transformation schemes ; inverse
operations ; powers of operations ; permutable opera-
tions ; similar operations ...... 1
4-7. Transformation group defined ; continuous group; infinite
group ; discontinuous group ; example of a mixed group 2
8. Identical transformation defined 3
9, 10. Examples of discontinuous groups ..... 4
11. Examples of infinite continuous groups .... 4
12. Definition of finite continuous group .... 5
13,14. Infinitesimal transformations; infinitesimal operators;
connexion between the finite and the infinitesimal
transformations of a group ; effective parameters ; ex-
ample 6
15. Independent infinitesimal transformations; independent
linear operators ; unconnected operators ... 7
16. The alternant (X1? X2) defined; the alternants of the
operators of a group dependent on those operators . 8
17. Verification of the relation between the infinitesimal
and finite transformations of a group .... 9
18. Simple isomorphism ; example 9
19-21. The parameter groups ; notation for summation . . 11
22. Transformation of a group ; type of a group ... 15
23, 24. Conjugate operations ; Abelian operations ; operations
admitted by a group ; sub-group ; conjugate sub-group ;
Abelian group ; special linear homogeneous group ;
sub-groups of the projective group of the straight line;
the translation group is the type of a group with one
parameter 16
25, 26. Order of a group; projective group of the plane; its sub-
groups ; another type of group ; projective group of
space ; some of its sub-groups ; similar groups . . 18
27. Euler's transformation formulae 20
28. A non-projective group ; number of types increases with
the number of variables 22
viii CONTENTS
CHAPTER II
ELEMENTARY ILLUSTRATIONS OF THE PRINCIPLE OF
EXTENDED POINT TRANSFORMATIONS
SECT. PAGE
29. Differential equations admitting known groups . . 23
30. The extended infinitesimal point transformation . . 23
31, 32. Differential invariants; example; some particular classes
of differential equations admit infinitesimal point
transformations ; examples ...... 25
33. Relation between the equations (— J +(v-) +(^l) =^
and dx2 + dy2 -f dz2 = 0 ; minimum curves ; Mongian
equations 28
34. Direct verification of this relation 29
35, 36. The infinitesimal operators admitted by lines of zero
length ; the conformal transformation group . . 31
CHAPTER III
THE GENERATION OF A GROUP FROM ITS INFINITESIMAL
TRANSFORMATIONS
37, 38. The parameters which define the identical transforma-
tion ; example illustrating method of finding these
parameters ; the symbol e m 34
39. A group not containing the identical transformation . 36
40. Method of obtaining the infinitesimal operators . . 36
41-43. First fundamental theorem ; the order of a group equals
the number of its operators ; illustrative examples . 38
44. The operators of the first parameter group ; preliminary
formulae and proof of the theorem that every finite
operation of a group may be generated by indefinite
repetition of an infinitesimal one ; transitive group ;
simply transitive group ; canonical equations of a
group ; simple relation between an operation and its
inverse when the group is in canonical form . . 41
45, 46. On finding the finite equations of a group when the
infinitesimal operators are given ; example ; the finite
equations of the group not often required ... 46
CONTENTS
IX
CHAPTER IV
THE CONDITIONS THAT A GIVEN SET OF LINEAR OPERATORS
MAY GENERATE A GROUP
SECT. PAGE
47. The fundamental set of operators of a group not unique ;
the structure constants ; the second fundamental theo-
rem stated and proved ; the converse stated ; illustra-
tive example 51
48, 49. Formal laws of combination of linear operators; examples 53
50, 51. Proof of converse of the second fundamental theorem . 57
52. Form in which any operation of the group can be ex-
pressed 59
53, 54. Examples 60
55. Reciprocal groups ; example 62
56, 57. Examples 62
58. The structure constants of a group the same as those of
its first parameter group 65
59. The converse of the first fundamental theorem . . 65
CHAPTER V
THE STRUCTURE CONSTANTS OF A GROUP
60. Jacobi's identity; relation between the structure constants
of a group ; the third fundamental theorem ; statement
of its converse 67
61. The structure constants vary with the choice of the
fundamental sets of operators; groups of the same
structure 68
62. The normal structure constants 70
j = k = n.
63. The group X; = 2 cjik xj t~ > example on group con-
struction 73
64. Proof of converse of third fundamental theorem ; example
on group construction ....... 74
65, 66. Solution of a system of differential equations ... 76
67. The three fundamental theorems 80
CONTENTS
CHAPTER VI
COMPLETE SYSTEMS OF DIFFERENTIAL EQUATIONS
SECT. PAGE
68. The terms unconnected and independent distinguished ;
complete system of operators ; functions admitting
infinitesimal transformations 81
69. The invariants of a complete system ; complete system in
normal form 83
70, 71. The form to which a complete system in normal form can
be reduced 84
72. The number of invariants of a complete system ; how
obtained 87
73. Integration operations of given order defined ; how the
order of the operations necessary for the solution of a
given equation is lowered, when the given equation is
a member of a given complete system ; an operator
which annihilates the invariants of a complete system
belongs to that system 87
CHAPTER VH
DIFFERENTIAL EQUATIONS ADMITTING KNOWN TRANS-
FORMATION GROUPS
74. Object of the chapter; formula for the transformation of
an operator to new variables ..... 90
75. Condition that a sub-group may be self-conjugate . . 91
76, 77. Condition that a complete system of equations may admit
an infinitesimal transformation ; second form of the
condition 93
78. Trivial transformations ; distinct transformations ; con-
dition that a system admitting x'=x + ttj may also
admit x'=x + tp£ 95
79. Reduced operators ; if a system admits any operators
it admits a complete system of operators ... 96
80. Properties of such a complete system ; it may give some
integrals of the given complete system of equations . 97
81. Form of the admitted operators necessary for further
advance towards solution of system .... 98
CONTENTS xi
SECT. PAGE
82. Restatement of problem at this stage ; origin of Lie's
group-theory 99
83. Simplification and further restatement .... 100
84. Maximum sub-group of group admitted ; new integrals . 101
85. If this sub-group is not self-conjugate further integrals
may be obtained without integration operations . . 101
86. Completion of the proposed problem .... 104
87. All the integrals can be obtained by quadratures when
the group is of a certain form 105
88-92. General remarks on the foregoing theory ; application to
examples ; points of special position ; theorem about
these points stated ; further examples .... 106
CHAPTER VIII
INVARIANT THEORY OF GROUPS
93-95. Second definition of transitivity ; invariants of intransi-
tive group ; geometrical interpretation ; cogredient
transformation; groups extended into point-pair groups 113
96, 97. The invariant theory of algebra in relation to group-
theory . . . .116
98-100. The functional form which in the more general invariant
theory takes the place of the quantic in the invariant
theory of algebra ; the invariant theory of this form ;
how the form may be obtained ; example . . . 119
101. Proof of theorem as to points of special position . . 124
102-104. Invariant equations with respect to a group; contracted
operators of the group with respect to these equations ;
proof of formula (X:/) = Xk.f; contracted operators
generate a group ; the order of the special points de-
fined by the invariant equations is equal to the number
of unconnected contracted operators .... 127
105, 106. Equations which admit the infinitesimal transformations
admit all the transformations of the group ; method of
obtaining such equations ; examples . . . .130
CHAPTER IX
PRIMITIVE AND STATIONARY GROUPS
107. Geometrical interpretation of the invariants of an in-
transitive group ; the contracted operators, with respect
to any invariant manifold of the group . . . 135
Xll
CONTENTS
SECT. PAGE
108-110. Primitive and imprimitive groups; the group which
transforms a manifold of an imprimitive group into
some other such manifold ; groups admitted by a com-
plete system of differential equations are imprimitive,
and conversely • 136
111, 112. The sub-group of a point; the operators of this sub-group;
conditions that the sub-groups of two points may be
coincident 139
113, 114. Stationary and non-stationary groups; analytical proof
that a stationary group is imprimitive .... 141
115. The functions <fe>, Ilyjt; structure functions ; stationary
functions 143
116. Simplification of stationary functions; the group Zx, ...,Zr
can be constructed when the structure constants and
stationaiy functions are assigned 144
117. Reduction of the operators of a group to standard form . 145
118. Integration operations necessary to find the finite equa-
tions of a stationary group 147
CHAPTER X
CONDITION THAT TWO GROUPS MAY BE SIMILAR.
RECIPROCAL GROUPS
119, 120. Necessary conditions for similarity; simplification of
these conditions preparatory to proving that they are
also sufficient 148
121. A correspondence between a g-fold in x space and ag-fold
in y space ; initial points ; the general correspondence
between the two spaces ; proof that it is a point-to-
point correspondence 151
122. Proof that the necessary conditions for similarity are
sufficient 153
123, 124. If s is the number of unconnected stationary functions
there are (n-s) unconnected operators Zx,..., Zn_s
permutable with each of the operators Xlt ..., Xr;
Zl, ...,Zn_g form a complete system whose structure
functions are invariants of Xlt ..., X, 154
125. If Xn ..., X,. is transitive, Zy, ..., Zn_s are the operators
of a group; if simply transitive, Z1, ...,Zn_a is also
simply transitive, and has the same structure constants
CONTENTS xiii
CHAPTER XI
ISOMORPHISM
SECT. PAGE
126, 127. The operators of the parameter groups in canonical
form expanded for a few terms in powers of the vari-
ables ; direct proof that the structure constants of a
group and its first parameter group are the same ; the
canonical form of a group not fixed till the funda-
mental set of operators is chosen 159
128. Two groups simply isomorphic when they have the same
parameter group 1G2
129, 130. When one group is multiply isomorphic with another; a
self-conjugate sub-group within the first corresponds to
the identical transformation in the second ; condition for
isomorphic relation between two groups ; simple groups 162
131, 132. When the structure constants of a group are given, the
structure constants of every group with which the first
is multiply isomorphic can be found ; the isomorphic
relation which may exist between the r independent
operators of a group, and the r non-independent opera-
tors of a group whose order is less than r 165
133. Examples of groups isomorphically related ; proof that
two transitive groups in the same number of variables
are similar, if they are simply isomorphic in such a
way that the sub-group of some point of general
position in the one corresponds to the sub-group of
some point of general position in the other . . .167
CHAPTER XII
ON THE CONSTRUCTION OF GROUPS WHOSE STRUCTURE CON-
STANTS AND STATIONARY FUNCTIONS ARE KNOWN
134. Object of the chapter 169
135. General relation between the structure functions of any
complete system of operators ; simplification of the
problem to be discussed 170
136. The system of simultaneous differential equations on
whose solution the problem depends . . . .171
137. Proof that this system is a consistent one; general method
of solution .173
138. Extension so as to apply to the case of intransitive groups 174
xiv CONTENTS
CHAPTER XIII
CONJUGATE SUB-GROUPS: THE CONSTRUCTION OF GROUPS
FROM THEIR STRUCTURE CONSTANTS
SECT. PAGE
139. A new set of fundamental operators Fx, ..., Yr is chosen
instead of the original set Xlf ..., X, 176
140, 141. Definition of the functions Hijk; proof that they are the
structure constants of Yl,..., Yr ; the functions ITy*;
identity connecting these functions . . . .177
142, 143. Definition of the operators Ilj, ..., n,. ; they form a group
with which Xlf ...,Xr is isomorphic; the equation
system Hq+i,q+j,k = 0 admits these operators . . 179
144,145. The equation system Hq+i,q+j,k = 0 defines sub-groups
of order r - q ; method of finding all such sub-groups ;
the group within which a given sub-group is invariant ;
the index of a sub-group 181
146. Method of finding all sub-groups conjugate to a given
sub-group 183
147. Method of finding all the different types of sub-groups . 186
148. Application of the preceding discussion to enable us to
determine the stationary functions of a group whose
structure constants are given 187
149-151. Illustrative examples ; a particular case of the general
theory 189
CHAPTER XIV
ON PFAFF'S EQUATION AND THE INTEGRALS OF PARTIAL
DIFFERENTIAL EQUATIONS
152. Element of space ; united elements; Pfaff's equation and
its solution ; Pfaffian system of any order ; generating
equations 194
153, 154. Alternant of two functions ; functions in involution ;
equations in involution ; homogeneous function sys-
tem ; necessary and sufficient conditions that n equa-
tions should form a Pfaffian system .... 196
155. Geometrical interpretation of solution of Pfaff's equation 201
156, 157. Lie's definition of an integral ; the problem involved in
the solution of a partial differential equation of the
first order 202
CONTENTS xv
SECT. PAGE
158. Proof that (u, v) = (u, v) 205
159. Proof that Pis not connected with Mj, ...,Mm . . . 206
160-163. On finding the complete integral of a given equation;
illustrative examples on the foregoing theory . . 208
CHAPTER XV
COMPLETE SYSTEMS OF HOMOGENEOUS FUNCTIONS
164. Necessary and sufficient conditions that a given system of
functions may be a homogeneous one .... 213
165. General definition of a complete homogeneous function
system ; structure functions of the system ; if all the
functions are of zero degree the system is in involution 214
166. If/ is annihilated by ul, ..., um, where «, , ...,um form a
homogeneous function system, Pf is also annihilated . 215
167. Proof of the identity (ic, (v, w)) + {iv, (u, v)) + (v, (w, u)) = 0;
the polar system ........ 216
168. The functions common to a system and its polar are
homogeneous and in involution thus forming an Abelian
sub-system ; satisfied system . . . . . .217
169, 170. Any complete homogeneous system is a sub-system within
a satisfied system ; complete systems of the same struc-
ture ; contracted operator of «j 218
171-174. The normal forms of complete homogeneous systems ;
systems of the same structure 220
175, 176. Every complete system of homogeneous functions is a sub-
system within a system of order 2 n ; two systems of
the same structure are sub-systems of the same struc-
ture within two systems of the same structure and of
order 2« 223
CHAPTER XVI
CONTACT TRANSFORMATIONS
177. Necessary and sufficient conditions that xt = Xj, p/= Pi
i =n i — n
should lead to zZ Pi dxi = Z^ Pi ^xi .... 226
178, 179. X1, ...,Xn, Pi, ...,P„ are unconnected; contact transfor-
mation defined ; geometrical interpretation ; the trans-
formation given when X1,..., Xn are given; example of
contact transformation 228
xvi CONTENTS
SECT. PAGE
180, 181. By a contact transformation a Pfaffian system of equa-
tions is transformed into a Pfaffian system ; examples
on the application of contact transformations to diffe-
rential equations 231
l-'J. I-".. Any two complete homogeneous systems of Functions of
the same structure, and in the same number of vari-
ables, can be transfoimied into one another by a homo-
geneous contact transformation; extension to the case
of non-complete systems of functions .... 234
184,185. Non-homogeneous form of Pfaff's equation; the corre-
sponding Pfaffian systems and contact transformations 238
186, 187. Example on the reduction of a function group to a simple
form by a contact transformation ; Ampere's equation
reducible to the form s = 0 if it admits two systems of
intermediary integrals 241
CHAPTER XVII
THE GEOMETRY OF CONTACT TRANSFORMATIONS
188. The generating equations of a contact transformation . 245
189, 190. Limitation on the form which generating equation can
assume ; interpretation 246
191. Contact transformation with a single generating equation 247
192. Special elements, and the special envelope . . . 249
193, 194. The three classes of element manifolds .... 250
195. Reciprocation 252
196, 197. Contact transformation with two generating equations . 253
198. Linear complexes 255
199, 200. Bilinear equations as generating equations, simplification 257
201, 202. The generating equations
x' + iy' + xz + z = 0, x {x' - iy') -y- z'=0;
to points in space x, y, z correspond minimum lines
in x', y, z ; to points in space x, y, z correspond lines of
a linear complex in x, y, z 259
203-205. To lines in space x, y, z, spheres in x', y', z ; to spheres in
x, y', z', a positive and a negative correspondent in
x, y, z; contact of spheres and intersection of lines ;
example 262
206-210. To a quadric in x, y, z a cyclide in x, y', z' ; to lines of
inflexion, lines of curvature ; further examples . . 265
CONTENTS xvii
SECT. PAGE
211-217. The generating equations
axx' + bijy + czz' + d = 0, xx' + yy' + zz' + 1 = 0
transform a point in one space to a line of the tetra-
hedral complex in the other ; a plane to a twisted
cubic ; a straight line to a quadric ; deduction of
geometrical theorem ; the generators in the quadric ;
case of degeneration ; illustrative examples on method 268
218. Point transformation 275
CHAPTER XVIII
INFINITESIMAL CONTACT TRANSFORMATIONS
219-221. Infinitesimal contact transformations ; characteristic
functions ; condition that an equation should admit
an infinitesimal contact transformation . . . 276
222, 223. Characteristic manifolds of an equation ; transformation
of, by a contact transformation ; geometrical interpre-
tation of infinitesimal contact transformation . . 278
224, 225. Linear element ; elementary integral cone ; Mongian
equations; correspondence between Mongian equations
and partial differential equations ; Mongian equations
and partial differential equation related to tetrahedral
complex 280
226, 227. The characteristic function of the alternant of two con-
tact operators ; transformation of operator by a given
contact transformation 284
228-232. Finite contact group ; extended point group ; its struc-
ture ; condition that two contact groups may be similar 286
233,234. Reducible contact groups; contact groups regarded as
point groups in space of higher dimensions . . . 292
CHAPTER XIX
THE EXTENDED INFINITESIMAL CONTACT TRANS-
FORMATIONS: APPLICATIONS TO GEOMETRY
235, 236. The transformation of the higher derivatives of z by an
infinitesimal contact transformation ; explicit forms
•»S-2 ™
237-239. The groups transforming straight lines into straight
lines, circles into circles 297
CAMPBELL j^
xviii CONTENTS
SECT. PAGE
240, 241. Transformations of the group of § 239; correspondence
between circles of the plane and lines of a linear com-
plex in space ; a projective group isomorphic with the
conformal group 302
242, 248. The twice extended contact operator in three variables ;
transformations admitted by s = 0 ; Ampere's equation 305
244, 245. The transformations which do not alter the length of
arcs on a given surface ; the measure of curvature
unaltered by such ........ 308
246-249. Surfaces over which a net can move ; geometrical treat-
ment of the question ; analytical discussion by aid of
Gaussian coordinates ; the group of movements of the
net 311
CHAPTER XX
DIFFERENTIAL INVARIANTS
250, 251. How to obtain the differential invariants of a given group 319
252. Differential invariants of the group x' = x, y'= % . 321
253, 254. Extended operators of projective group of the plane ;
invariant differential equations ; absolute differential
invariants ; the invariants of lowest order of this group 322
255-257. The group of movements in non-Euclidean space ; ex-
tended operators of; differential invariants of; geo-
metrical considerations help in determination of . . 326
CHAPTER XXI
THE GROUPS OF THE STRAIGHT LINE, AND THE
PRIMITIVE GROUPS OF THE PLANE
258. The possible types of groups in a given number of variables 331
259. Operators arranged in systems according to degree of the
coefficients in the variables 332
260. The possible types of groups in a single variable . . 333
261, 262. Simplification of any operator of the linear homogeneous
group 335
CONTENTS xix
SECT. PAQE
263, 264. The possible types of linear homogeneous groups in the
plane 339
265-270. The primitive groups of the plane ; operators of the first
degree ; the group cannot have operators of the third
degree ; possible form of operators of the second degree ;
structure constants of the group ; possible types of . 342
CHAPTER XXII
THE IMPRIMITIVE GROUPS OF THE PLANE
271. Can be arranged in four classes, and thus successively
found ....
353
354
357
362
364
272-274. The groups of the first class
275-279. The groups of the second class .
280, 281. The groups of the third class .
282. The groups of the fourth class .
283. The systems of curves which are invariant for the different
types of imprimitive groups 365
284. Enumeration of the mutually exclusive types of imprimi-
tive groups of the plane 368
CHAPTER XXIII
THE IRREDUCIBLE CONTACT TRANSFORMATION GROUPS
OF THE PLANE
285-287. Condition for the reducibility of a system of contact opera-
tors of the plane ; an irreducible group of the plane is
a transitive group of space ; the form of the operators
of the first degree 370
288-290. The irreducible groups in the first class have six indepen-
dent operators ; the structure of any such group . . 373
291, 292. Every group in this class is of the same type . . . 377
293, 294. The remaining irreducible contact groups of the plane . 378
CHAPTER XXIV
THE PRIMITIVE GROUPS OF SPACE
295, 296. The curves which admit two infinitesimal projective
transformations must be straight lines or conies . . 381
297. Any sub-group of the projective group must leave unaltered
either a point, a line, or a conic 383
XX
CONTENTS
SECT. PAGE
298. A projective group isomorphic with the group of the
origin ; the cases when this projective group has no
invariant . 385
299, 300. The case when it has as invariant a straight line . . 386
301-305. The cases when it has as invariant a conic . . . 389
306. Enumeration of the types of primitive groups . . . 396
CHAPTER XXV
SOME LINEAR GROUPS CONNECTED WITH HIGHER
COMPLEX NUMBERS
307-309. Properties of simply transitive groups which involve the
variables and the parameters linearly in their finite
equations 398
310, 311. Determination of all the groups of this class in three
variables ......... 401
312. The theory of higher complex numbers .... 406
313-315. To every such system a group of the class considered will
correspond, and conversely. Examples . . . 408
INDEX 411
ERRATUM
Page 62, line 14, for Ys read Yr
CHAPTER I
DEFINITIONS AND SIMPLE EXAMPLES OF GROUPS
\
§ 1. If we have two sets of variables, xli...,xtt and xx\ . . ., xn',
connected by the equations
(1) x^ = ji(x1, ..., xn), [i=l, ...,n),
they will define a transformation scheme, provided that we can
solve the equations so as to express the variables xx, ..., xn in
terms of the variables xx, ..., xn'.
We shall denote the transformation scheme (1) by S.
The operation, which consists in substituting for xx, ...,xn
in any function of these variables fv ...,fn respectively, will
be denoted by Sx, or simply by S when there is no need to
indicate the objects on which the operation S is performed.
So Sy will denote the operation of substituting for yv ...,yn
respectively, fx (yv . . . , yn), ...,fn (yls .. ., yn) respectively.
Similarly the operation which consists in substituting for
Xj the function /^(/j, ...,/„) will be denoted by S2, and so on.
Solving the equations (1) we obtain the algebraicaUy
equivalent set
(2) x{ = Fi «, . . . , xn'), (i = 1 , . . . , n).
From (1) and (2) we see that
We therefore denote the scheme (2) by 8~\ and the operation
af substituting F1(x1, ..., xn), ..., Fn(x1, ...,xn) for xv ...,xn
respectively by jS^-1.
The two schemes (1) and (2) are said to be inverse to one
another.
§ 2. If we have a second transformation scheme T, viz.
xi- 4>%{xi> •••»«»)> (* = *i .»»»).
jhen TSX will denote the operation of substituting fi (#19 ..., (j)n)
for x^
CAMPBELL '« £ g
2 GENERAL DEFINITION OF A GROUP [2
The function ft (4>x, ..., <pn) may be more compactly written
fi<t>, the function /;(</>! (^x, ...,^n), ...,0n(^1,...,^J) may be
written fi^^r, and so on.
In TS the order in which the operations are to be taken
is from right to left ; but it should be noticed that, / being
the functional symbol which corresponds to S, and <p the
functional symbol which corresponds to T, the functional
symbol which corresponds to TS is not <£/but/<£.
So if we have a third transformation scheme U, viz.
xi = Yi \xn •••? xn),> \l = *> •••» nh
UTS would denote the operation which consists in first opera-
ting with S, then operating with T on this result, and finally
operating with U \ the functional symbol which corresponds
to UTS is f<t>if' : that is, UTS is the operation which consists
in substituting ^(py, ...,fn(f>\j/ for xv ...,xn respectively*.
ST denotes the operation of substituting $,/, ..., $n/ for
xi,...,xn respectively, and TS the operation of substituting
for xv ..., xn respectively, fi<t>,---,fn<t>; if then
M = <l>if> (* = i>— »»)i
ST = TS, and the operations S and T are said to be per-
rnutable.
§ 3. In accordance with what precedes, STS^ denotes the
operation of replacing xi by F,L <pf ; it fuliows therefore that
when STSx~l is applied to fi(x1, ...,xn) this function becomes
fiF<t>f; that is, since ftF= x{, it becomes <^.(/l5 ...,/„).
We thus see that the operation STS'1 has the same effect
on the variables x{, ..., xn', when expressed in terms of xlt . . . , xn
by the scheme S, viz.
%i = Ji \xl> • ■ • j xn» K1 = *» • ■ • j n)>
as the operation Tx> has on the variables x{, . . . , xn' ; STS~X is
therefore said to be an operation similar to T with respect
to S.
§ 4. If we have a system of transformation schemes S1} S0, . . . ,
and if the resultant operation generated by successively per-
forming any two operations of the system is itself an operation
of the system, then the transformation schemes are said to :
form a group.
* In Burnside's Theory of Groups the order of operations is taken from left to
right. The reason why we have adopted the opposite convention is that we
shall deal chiefly with differential operators, and it would violate common
d d
usage to write — y in the form y — .
dx * dx
8] CONTINUOUS AND DISCONTINUOUS GROUPS 3
§ 5. A group is said to be continuous when, if we take any
two operations of the group S and T, we can always find
a series of operations within the group, of which the effect
of the first of the series differs infinitesimally from the effect
of S ; the effect of the second differs infinitesimally from the
effect of the first ; the third from the second and so on ; and,
finally, the effect of the last of the series differs infinitesimally
from T. Naturally this series must contain an infinite number
of operations unless S and T should themselves chance to differ
only infinitesimally.
§ 6. If the equations which define the transformation
schemes Sv S2, ... of a group involve arbitrary functional sym-
bols the group is said to be an infinite group ; but we shall
see that a group, with an infinite number of operations within
it, is not necessarily an infinite group.
§ 7. A group is said to be discontinuous if it contains no
two operations whose effects differ only infinitesimally.
It should be noticed that the two classes of continuous
and discontinuous groups, though mutually exclusive, do not
exhaust all possible classes of transformation groups.
An example of a transformation group which belongs to
neither of the above classes is
x' '— c»x + a,
where a is a parameter and w any root of xm = 1 .
A series of transformations within the group, the effects of
consecutive members of which only differ infinitesimally, could
be placed between
x' = (ox + a and xf = a>x + b,
, , b — a , 2 (b — a)
viz. x = a>x + a-] > x = cox + a + — '-,-"■>
n n
, n — 1 ,, x
x — <x)X + a H (b — a),
n v J
where n is a very large integer ; but such a series could not
be placed between
x'= oox + a and x' = a>'x + b
if to and a/ are different mth roots of unity.
§ 8. The transformation scheme
is called the identical transformation ; if it is included in the
transformations of a group, the group is said to contain the
identical transformation.
B 2
4 EXAMPLES OF GROUPS [9
§ 9. A simple example of a discontinuous group is the set
of six transformations,
1 , x—l , 1 , , , x
x'=x, x' = , x'- 1 a =-, x- l-x, x = -
1— x' x x' x—l
by which the six anharmonic ratios of four collinear points
are interchanged amongst themselves.
If we denote the six corresponding operations by S1 (which
is equal to unity since it transforms x into x), S2, Sz, #4,
S5, >S'6 respectively, we verify the statement that these opera-
tions form a group when we prove that S.2S3 = 8X, Si S5 = S3,
and so on.
Inversion with respect to a fixed circle offers an even
simpler example of a discontinuous group ; it only contains
two operations, viz. the identical operation #, and the opera-
ct x
tion 82 which consists in replacing x by —^ -2 and y by
a2y _ # x +y
—: — Z-? when the circle of inversion is x2 + y2 = a2.
x'2 + yz *
The group property follows from the fact that $22 = Sv
§ 10. In the above two examples there are only a finite
number of operations in the gi-oup ; the set of transformations,
x'—ax + ^y, y'=yx + by,
where a, j3, y, 8 are any positive integers, is an example of
a discontinuous group with an infinite number of operations.
The group property follows from the fact that from
x/ = ax + /3y, y' = yx + hy,
and x" — px' + qy\ y" = rx' + sy',
where p, q, r, s are another set of integers, we can deduce
x"= (pa + qy)x + ('p(3 + qb)y, y" = (ra + sy)x + (rp + sh)y,
where the coefficients of x and y are still positive integers.
§ 11. Simple examples of continuous groups are the fol-
lowing :
(i) x'=f(x), y'=4>(y) I
where / and <p are arbitrary functional symbols ; the group
property follows from the fact that these equations and
where X and \x are other arbitrary functional symbols, lead to
x"=\f(x), y"=ixcp(y).
12] FINITE CONTINUOUS GROUPS 5
(2) x'=f(x, y), yf= <f>(x, y), z'= y{z)
where/, 4>, and ^ are all arbitrary functional symbols.
(3) af=f(z,y), y'=<p{x,y)
where/ and <£ are conjugate functions ; for if 6 and \}r are two
other coujugate functions, and
x"=Q(x\y'\ y"=^(x\y%
then x" + iy" = F (x' + iy') =F$>(x + iy),
so that x" and y" are also conjugate functions of x and y ;
that is, the transformation system, which is obviously con-
tinuous, has the group property.
(4) x'=f{x,y,z), y'=^(x,y,z), z'=f{x,y,z)
where /, (/>, \jr are functions of their arguments such that their
Jacobian »(/,»,*) _ j
d (x, y, z)
The group property follows from the identity
J) (a?, y, z) d («', 2/', z') d (x, y, z) '
These are examples of infinite continuous groups, for the
transformation schemes in (1), (2), (3), (4) involve arbitrary
functional symbols.
§ 12. If the transformation scheme
x% z= ft \Xi, ••• ? xn, <Xj, ..., Ozrj, [i = i , .,,, n)
defines a group ; that is, if from the equations
X% = fi\X^, ...,Xn, ttj, .. ., &rj,
X{ — Ji \X± j • • • ? xn , Oj, . . . , or J
we can deduce x// = fi(x1, ...,xn, cv...,cr),
where av . . . , ar and b1,...,br are two sets of r unconnected arbi-
trary constants, and cv ..., cr are constants connected with these
two sets, then this group is said to he finite and continuous.
If values of av ..., ar can be found such that
X{ =Ji \Xd •••? xn, ctj, ...,oir), \i = i,...,n)
the group contains the identical transformation ; if a^, ...,ar°
are these values, a^0, ..., ar° are said to be the parameters of the
identical transformation. Finite continuous groups do exist
which do not contain the identical transformation, but the
properties of such groups will not be investigated here.
6 THE INFINITESIMAL TRANSFORMATION [13
§ 13. A transformation whose effect differs infinitesimally
from the identical transformation is said to be an infi-
nitfsimat transformation. The general form of such a
transformation is
%i == &'i + c f j- \xv . . . , xn), yi = 1 j . . . , n )
where t is a constant so small that its square may be neglected.
If </> (xv ..., 05n) is any function of xv ...,xn, then if we expand
0 (a;/, ...,#/) in powers of t, neglecting terms of the order t2,
we get
0(a*'» ...»«»') = ^(ai + ^i»...»a'» + ^»)
If then we let X denote the linear operator,
<j>(xv...,xn) + t(^r + ... +£nj£ )
Ci \xv • • • j #m) > „ +••■+ few (#!>••• j #«)
3^ *»™ "' "' 3a;„
0 (a>/, . . . , %n') = (l+tX)(t)(xv..., xn)>
so that we take 1 + tX
to be the symbol of an infinitesimal transformation ; and we
call X the infinitesimal operator, or simply the operator, which
corresponds to this infinitesimal transformation.
We shall see that any transformation whatever of a finite
continuous group which contains the identical transformation
can be obtained by indefinite repetition of an infinitesimal
operation ; that is we shall prove that if
X} = j % \xv . . . , xn, ttj, . . . , ar), ( i = l , . . . , n)
are the equations of such a group,
z X \m
fi(xv ...,xn, av ...,ar) = the limit of (l H J xit
when m is made infinite, and X is some linear operator.
This limit is, we know by ordinary algebra,
(1 + hx+hX2+hX3+~)
x^.
§ 14. A simple example of a finite continuous group is the
projective transformation of the straight line
/ (Jj-\ QO ~f~ (Xn
X — -
a^x + a^
where av a2, a3, a4 are four arbitrary constants ; the group
15] INDEPENDENT OPERATORS 7
property of these transformation schemes can be easily
verified.
In this group four arbitrary constants appear, but only
three effective parameters, viz. the ratios of these constants ;
it is always to be understood that the parameters of a group
are taken to be effective ; thus, if ax and a2 always occurred
in the combination ax + a2 they would be replaced by the
single effective parameter ax.
The identical transformation in the above projective group
is found by taking the parameters a2 = «3 = 0 and ax = a4 .
If we take a1 = a4(l+e2), a2 = e1ai, az— — e3a4, where
ex, e2, e3 are small constants whose squares may be neglected,
, ( 1 + e2) x + ex } 2
1— ezx l
This is the general form of an infinitesimal transformation of
the projective group of the straight line.
§15. If xi' = xi + ek£hi(x1,...,xn)i (kZit["[r)
are a set of r infinitesimal transformations, they are said to be
independent if no set of r constants, \v ..., Kr, not all zero, can
be found such that
\£li+ '" +K€ri = °> (*= !»•••!*)•
The r linear operators, Xv ...,Xr, where
are said to be independent when no r constants, \v ...,Ar, not
all zero, can be found such that
Aj-STj + ... +Ar.A.r = 0.
Any linear operator which can be expressed in the form
AjAx+ ... +\rXr
is said to be dependent on Xv ...,Xr.
If we have r operators, Xv ...,Xr, such that no identical
relation of the form
ylr1X1+ ...+\lsrXr = 0
connects them, where ^x, . . . , ^r are r functions of the variables
xv . . . , xn, not all zero, they are said to be unconnected operators.
It is necessary to distinguish between independent operators
and unconnected operators ; unconnected operators are neces-
8 DEFINITION OF THE ALTERNANT [15
sarily independent, but independent operators are not neces-
sarily unconnected ; thus
ox' oy' oz
are unconnected operators, but X, Y, Z where
X = v z — ■> Y=z- x — , Zi — Xz 2/ —
ycz cy ox oz' oy * ox
are three connected operators, since
xX + yY+zZ=0,
and yet they are independent.
In the projective group of the straight line there are three
independent operators, viz.
- — j X r 5 ££ ^ )
Sx ox ox
but only one unconnected operator.
We shall find that there are always just as many indepen-
dent operators in a group as there are effective parameters.
§ 16. If X1 and X2 are any two linear operators, the symbol
Xx X2 means that we are first to operate with X2 and then
with Z2 ; the symbol Xx X2 is not then itself a linear operator ;
but X1X2 — X2X1 is such an operator, since the parts in Xx X2
o2
and X0 X, which involve such terms as - — - — , are the same
in both. oxi*x*
The expression X1X2-X2Xl is written (Xv X2) and is
called the alternant of Xx and X2.
In the projective group of the straight line we see that
(-
Ss o
X—) = r~,
cx' ox
dec' ore
( c
V ax
Z2 — - ) = X1 — j
Sax drc
so that the alternant of any two of the three infinitesimal
operators of the group is dependent on these three operators.
This will be proved to be a general property of the infinitesimal
operators of any finite continuous group.
18] ISOMORPHISM 9
§ 17. The most general infinitesimal operator of the pro-
jective group of the straight line is X where
d
X={ei + e2x + ezx2)-^
and ev e2, e3 are arbitrary constants.
If we take
(1). V =_2(4e1e3-e22)-2 tan"1 {(le^-e*)-* (2ezx + e.?)\,
it is easily verified that
and therefore
is equal to
1 eZ 1 cZ2 v , /^~ e22 Vu^-e* e2 x
l + nc72/ + 2!^+-Mv^~^? — 2 — V~W'
and this by Taylor's theorem is equal to
V
e3 4e32 2 w ' 2e3
If we substitute for 2/ its value in terms of x we shall have
an expression of the form
where a1} a2,a3, a4 are functions of e1> e2, e3; and we thus verify,
for the case of the projective group of the straight line, the
general theorem that any transformation of a group can be
obtained by repeating indefinitely a properly chosen infini-
tesimal transformation.
§ 18. If we have two groups
and y/=<l>i{yv
where m and n are integers not necessarily equal ; and if we
have a correspondence between S^, ...,ar the operations of the
first, and Tai, ...,ar the operations of the second such that to
every operation Sav ...,ar a single operation Tav ...,ar corre-
sponds, and to every operation Tav ...,ar a single operation
, Xn, ttj, . . ,, C£j.J,
(i = 1, ...,ri)
' y 7ft' i> • • • ' r/»
(i = l,...,m)
10 ISOMORPHISM [18
Sav...,Or and to the product Sav...,ar Sbv...,br the product
2\jv ...,ar Tbv •••, br> then the two groups are said to be simply
isomorphic.
It might appear at first that any two groups with the same
parameters would be simply isomorphic; we could of course say
that Salt . . . , ar corresponds uniquely to Tav ...,ar and Sbv ...,br
to T(jv ...,br, but it would not follow that Sai, ...,ar Sbi, ••■ibr
corresponded to Tav ...,ar Tfa, ..., br> For from the definition
of the group
£>ai> •••> ar £>bi> • ••> br = ben •••» Cr>
where cv...,cr are functions of the two sets al,...,ar and
61, ...,br; and these functions will naturally depend upon the
forms of the functions fv ...,fn which defined the first group ;
while from T Tl j T
J-ai> •••}«}• J-(Jxi •••5 or — ^"/v •••> "Yri
where yv ...,yr are functions of av...,ar and bv...,br, whose
forms depend on the forms of the functions <f>v ..., $m, we could
not in general conclude that yx = cv ...,yr = cr unless the two
groups are specially related.
An example of two simply isomorphic groups is offered by
Jb*i — — Cvi JO] ""T C6-i Cvo tv,> 3 t£o ~~~ tX-i *^o
and y{ = yx + a2y2 + log ax , y2' = y2.
If we take two operations of the first
*t/j — — Cv-i tbi i~ Cc-i C6p <-C'o ) tb'n ^— IX-i \Aja •
JC-* -^ C/-J \h-\ "J" O-i UtypCn , kCo "~~ ^1 ^o J
we deduce xx — c1x1 + c1c2x2, x2" = cxx2 ,
where c1 = a1bv c2 = a2 + b2,
so that the group property of the first is verified.
Taking two operations of the second
Vi =Vi+ a2y-2 + log ax , y2' = y2 ,
v" = Vi + \vl + log h > y2" = yz>
we also deduce
V" =2/1 + ^2/2 + iogcx , y2" = y,
where cx = a2 bx, c2=a2 + b2,
and thus verify the group property of the second and its
simple isomorphism with the first.
19] THE PARAMETER GROUPS 11
§ 19. Returning now to the definition of a finite continuous
group and writing fa (xx, ...,xn,ax, ...,ar) in the abridged form
fa(x, a) we see that if
xi = fi («• a\ xi" = fi « &)>
then Xi"=fi(x,c),
where ck = <j)k(av ...,ar, b}, ...,br), (k=l,...,r).
It will now be proved that these functions <f>v ...,$,. define
two groups, one of which is simply isomorphic with the given
group.
It is to be assumed that fa is an analytic function of
xv ..., xn, av .. ., ar within the region of the arguments xv ..., xn
av ...,ar; and also that the parameters are effective ; that is
if we suppose fa expanded in powers of xv ..., xn the coefficients
will be analytic functions of av ...,ar, and there will be exactly
r such functionally unconnected coefficients in terms of which
all other coefficients can be expressed.
From the group definition we have
fa (x, c) = x{' = fi (xf, b) = ^ (/x (x,a),...,fn (x, a),bv...,br),
and since the parameters are effective we have
0) cft = 0ftK»«-Jar» K— A)> (&=l,...,r).
Also x{ = Fi (x\ a), (i = 1 , . . ., n)
being the inverse transformation scheme to
we have
fi K b) = fi (x, c) = fi (Fx (a/, a),...,Fn (x', a), cv..., cr) ;
and therefore if we expand fa {x\ b) in powers and products of
as/, . . . , xn', since there are exactly r parameters involved, we
see that in the expansion of
(2) fi (Fi « o)...Fn (x\ a), cv . . . , cr)
there must be exactly r unconnected coefficients.
We further see that bk can in general be expressed in terms
of av . . . , ar, cv...,cr subject to certain limitations in the values
which av...,ar, cl,...,cr can assume in order that (2) may
remain an analytic function of its arguments.
Thus suppose we have the equations
f(xt y) = a, <t> (x, y) = j3,
a necessary condition that we may be able to express x and
y in terms of a, j3 is that the Jacobian of the functions / (x, y)
12
THE PARAMETER GROUPS
[19
and ^>{x,y) should not vanish identically, or as we shall say
the functions must be unconnected. The form of the functions
/ and <f> may, however, be such that whatever the values of
x and y, real or complex,/ cannot exceed an assigned value a,
nor (f> an assigned value b ; the equations
f(x,y)=a, 4>(x,y) = p
could not then be solved unless a ^ a and ft *^b.
When we come to seek the conditions that a group may
contain the identical transformation we shall have to make
ak = Cfr, and the result may be that we cannot solve the equa-
tions (1), and in this case the group will not contain the
identical transformation.
In general, however, we can express b^ in terms of av ..., ar,
v
,6'
,, and therefore in the equations
ck = <l>k(ai>—>ar> K-">K)> (k~ 1>~->r)
the functional forms <f>l, ..., </>r are such that the determinant
*4>i Hi
d&!
n.
Hr
cannot vanish identically.
Similarly from x// = fi (x', b) we deduce x(— F{ (x", b) ; and
from x/ = fi(x,a) and from these identities we have
ft (x, a) = F{ {x", b) = F{ (/, (x, c), . . ., /„ (x, c), &,, . . . , br) ;
so that we see that a^ can be expressed in terms of 6l5 ..., br,
'!■>
c„ and conclude that the determinant
Hi
Hr_
cannot vanish identically.
We can therefore conclude that the equations
(3) yjc = <f>k(Vi> -»2/f. «i,...,«r), {k = l,...,r)
define a transformation scheme with r effective parameters,
20] THE PARAMETER GROUPS 13
and we shall now prove that these are the equations of
a group.
We have f4 (a/, b) - x{'= ft (x, c) =f{ (x, <f> (a, b)) ;
and if we take any other set of parameters yv ...,yr,
<"=fi W> y) = fi K <t> (&, y)) = fi (x, <f> (a, $ (b, y))).
Now fi {x", y) - fi (x, $ (c, y)) = ^ (x, <£ ((/> (a, b), y)),
so that by equating the coefficients in these two expressions
for fi {x"y) we have the identity
<$>k (a> 4* (&> y)) = 4>k (<£ K b)> y)-
This identity leads at once to the group property of (3), for
by its aid we deduce from
yk = $k(y>a) and yk'=<i>Ti{y,>h) = (Pk((i>{y>a)>h)
that y^ = (f)k(y,(f)(a,b)),
that is the equations (3) generate a group which is known as
the first parameter group of
Xj —ji\X}_, ...,xu, a1,...,arj, (i = i,,,.,?i).
It is an obvious property of this parameter group to be its
own parameter group.
From the definition of simple isomorphism we see that two
groups are then, and only then, simply isomorphic when
they have the same parameter group ; the first parameter
group is therefore simply isomorphic with the group of which
it is the first parameter group.
§ 20. In exactly the same way we see that the equations
yk=<t>k(av~>ar> yi>-'->yr)> (fc = i,...,r)
are the equations of a group.
This group is called the second parameter group ; it is its
own second parameter group ; but it is not isomorphic with
the original group ; for from yh'= <t>h{a, y), yh"= $k{b, yf) we
deduce y{' ■= <pk (c, y), where ch = ^^.(615 ..., br, ax, ..., ar), and
(f)k (b, a) is not generally equal to (f>k (a, b).
The two parameter groups are such that any operation of
the first is permutable with any operation of the second.
This comes at once from the fundamental identity
4>k («' <t> (b> c)) = <Pk (# ia> h)> c)>
which is true for all values of the suffix k and the arbitrary
parameters a1, ..., ar, blt ..., br, cv ..., cr ; for to prove that
yk=<t>k{y>a) and yk'= 4>k(b,y)
14 NOTATION FOR SUMMATION [20
are permutable operations it is only necessary to prove that
4>h (<t> (h y), a) = <j>k (b, <p (y, a)).
§ 21. As an example we shall find the first parameter group
of the general linear homogeneous group,
the summation being for all positive integral values of h from
1 to n inclusive.
As such summations will very frequently occur it is neces-
sary to employ certain conventions to express them. The
subscripts will always denote positive integers ; those which
vary in the summation will be supposed to go through all
positive integral values between their respective limits, thus in
where the summation is for all positive integral values of
a from p to r inclusive, and for all positive integral values
of /3 from q to k inclusive, we should indicate the sum by
o--V, 0 = 3
When the two limits are the same we should write the above
sum in the form
a = j5 = h
2* Ca.pjh#i Kits'
This would not of course mean that a = /3 throughout the sum-
mation ; a summation in which a = (3 would be expressed by
o = k
j£i Caaj Aaj Aa£.
a = p
When the lower limit is unity it will be omitted, thus when
p = 1 the sum would be written
Ji, Capj^pi^-ak-
Expressing the linear group in this notation from
h = n h = n
xi="LaMxh and Xi"=^bhixh',
22] SIMILAR GROUPS 15
h = n
we obtain xj'—jLcHxhr
where Chi=^ahkhi'
If then yhi, ... are n2 variables, the linear group
k = n
yhi=^akiVhk
is the first parameter group of the general linear homogeneous
group in n variables.
It will be noticed that this group is itself a linear homo-
geneous group in n2 variables, but it is of course not the
general linear group in n2 variables.
The second parameter group is
k = n
§ 22. If in any given group
(1) xif=fi((c1,...,xn, av... ,ar), (i=l,...,n)
we pass to a new set of variables yv ...,yn where
(2) 2/i = 9i {xV"i xn)i
and to a cogredient set y/, ...,yn' given by
V% = 9i\xi "'b/j
where gi,...,gn are any n unconnected functions of their argu-
ments, we must obtain equations of the form
(3) yj = <l>i(yli...,yn,a1>—>arl (*=i,...,»).
We are now going to find the relation between the two trans-
formation schemes (1) and (3).
Let T denote the operation which replaces x^ by gl , x.2 by g2 ,
and so on.
If then «» =<%(&> •••>#«)
is the inverse scheme to (2), T'1 will denote the operation
which replaces ^ by (r^.
We now take Sa to be the operation which replaces x{ by
fi(x, a) and Sh the operation which replaces xi hyfi(x, b).
The operation TSaT'1 acting on yi that is on <ft(ff15 ...,xn)
will transform it into y{ ; for
16 CONJUGATE OPERATIONS [22
TSaT-*gi(xv...,xn) = T8agi(Glt...,Gn) = TSax{,
and TSax4 = Tft{xx, ...,xn, aly ...,ar) =fi(g1,...,gn, av...,ar),
and fi(gl,...,gn, av...,ar)=fi(yl,...,yn, alt ...,ar) = y(.
The operations of the transformation schemes (3) are
therefore Z^ T~\ TSh T~\ . . .
and since TSaT~l TShT~l = TS^T'1 = TSeT~\
we see that the equations (3) are the equations of a group
simply isomorphic with the group (l). The two groups (1)
and (3) are said to be similar. Similar groups are therefore
simply isomorphic, but it is not true conversely that all
simply isomorphic groups are similar. The necessary and
sufficient conditions for the similarity of groups are obtained
in Chapter X. It will then be seen why it is not possible to
transform the two isomorphic groups given in § 18 into one
another. Groups which are similar are also said to be of the
same type.
§ 23. It will be proved later that groups which contain the
identical transformation can have their operations arranged
in pairs which are inverse to one another ; that is to every
transformation Sa another transformation Sh of the group will
correspond in such a way that the product of the two will be
the identical transformation. If then T is any operation
within the group, T~l will also be an operation of the group,
and so will the operation TST'1. This operation is said to be
conjugate to S with respect to T; if TST~X is equal to S,
whatever operation of the group T may be, then 8 is per-
mutable with every operation of the group and is said to be
an Abelian operation.
If T is an operation of the group so is TST~X ; but even if
T is not such an operation, T8T~X may be an operation of the
given group : we should then say that T was an operation
which transformed the group into itself.
If Tx and T2 are two operations each of which transforms
a given group into itself, then T-^iST^1 is an operation within
the group ; T2 Tx ST^1 T.r1 must then be within the group ;
that is, since 'T^1 T2~l = (T2 TJ-1, T, T} is also an operation
which transforms the given group into itself.
It follows therefore that the totality of operations with the
property of transforming the group into itself, or as we shall
say the totality of operations which the group admits, form
a group. This group, however, need not be finite.
24]
SELF-CONJUGATE SUB-GROUP
17
§ 24. If out of all the operations of a group a set be taken
not including all the operations of the group, this set may
itself satisfy the group condition ; in this case it is said to be
a sub-group of the given group.
Let Slf ^2, ..., Ta, T2, ... be the operations of a group, and
suppose that Sv S2, ... form a sub-group, then Th8lTlt~1^
T^S^T^-1, ... which (§ 22) is a similar group to Sv S2, ... is
said to be conjugate to the sub-group SVS2,.... Sub-groups
which are conjugate to one another are also said to be of the
same type.
If, whatever the operation Tk may be within the group
Slf S2, ..., Tlt Tv ... the sub-group Tk 8, Th~\ Tk S2 Tf\ ...
coincides with Sv S2, ..., then the sub-group Sv S2, ... is said
to be a self-conjugate sub-group. It will be noticed that it
is not necessary in order that the sub-group may be self-
conjugate, that Th Sh Tfr'1 should be identical with 8h, but
only that it should be some operation of the system Sv S2, ....
A group such that all its operations are commutative is called
an Abelian group.
It is easily proved that if a group contains Abelian opera-
tions they form an Abelian sub-group.
Example. The linear homogeneous transformation schemes
h = n
X
l=^<ahiXh> (*=l,....,»)i
where the parameters are subject to the single condition
a
ii
a
m
a
n\
a
nn
= 1,
form a group with (n2 — 1) effective parameters.
If Sa is a transformation included in this scheme, and Ma
the above determinant, then, Sh being any other transformation
of the scheme and Mh the determinant which corresponds to it,
the determinant of SaSh is MaMh; and therefore, since this is
unity, the transformations generate a group. This group is
called the special linear homogeneous group; it is a sub-
group of the general linear homogeneous group. It is also
self-conjugate within it; for if T is any operation of the
general group, the determinant of TSaT~l is the same as that
of £a, and therefore TSaT~l is itself an operation of the
special linear group.
CAMPBELL
c
18 PROJECTIVE GROUP OF THE PLANE [24
Example. The projective group of the straight line
" a3x + a4
contains the sub-group
x' = a1x + a2.
This sub-group contains two sub-groups, viz.
x' = ax and x' = x + a;
the first is the homogeneous linear group, and the second is the
translation group.
We shall prove later that these are the only types of finite
continuous groups of the straight line ; that is, all other
groups of the straight line are transformable to one of these
by the method of § 22 ; it will also be proved that every
group which contains only one parameter is of the type
x' = x + a,
that is, the type of the translation group of the straight line.
§ 25. A group which contains r effective parameters is said
to be of order r, or to be an r-fold group. We now write
down some groups of transformations of the plane.
The eight-fold projective group is
, an x + a2l y + a31 , a12 x + a22 y + a32
x = > y — ■ •
a13x + a23y + a33 aisX-\-a23y + a33
The identical transformation is obtained by taking
®11 = ^22 = *^33'
and making the other parameters zero ; the eight infinitesimal
operators (§ 13) are then found to be
d d d d d d
r— 3 r— > X— ) ?/-) X—> Vr-I
ox cy dx oy °y °X
0T - h XV r— j XV- h^/V*
Ix b dy J<ix J ly
The projective group has as a sub-group the general linear
group, viz.
x'= anx + a21y + a3X, y' ' = a12x + a22y + a32,
of which the infinitesimal operators are
a a a a a a
t— » r— > 05 r— > Ur~J # — > « — •
d# d?/ d£ d2/ OW d#
One sub-group of the general linear group is the group of
movements of a rigid lamina in a plane, viz.
26] A NON-PROJECTIVE GROUP 19
x' = x cos 0 + y sin 0 + a: , y' — — x sin 0 + y cos 0 + a2i
a1} a2, and 0 being the arbitrary parameters.
The identical transformation is obtained by putting
ax = a2 = 0 = 0,
and the infinitesimal transformations by taking alf a2, 0 to be
small unconnected constants ; the infinitesimal operators are
d d }> d
^— > ^— i v^ — #^— •
da t>2/ ^ ty
Each of these sub-groups could be obtained from the pro-
jective group by connecting the parameters of the latter by
certain equations ; thus the general linear group was obtained
by taking al3 = a2z = 0. It must not, however, be supposed
that if we are given a group, and connect its parameters by
some arbitrarily chosen equation, the resulting transformation
system will generally be a sub-group ; this would only be true
for equations of a particular form connecting the parameters of
the given group.
It has been stated that there are no groups of the straight
line which are not types of the projective group of the line, or
of one of its sub-groups. In space of more than one dimen-
sion, however, groups do exist which are not of the projective
type ; thus in the plane the equations
, a,x + a9 . yxr + anXr + aRxr~1 + ... + ar+,
af=— -» y — — ; — r-r — j
^x + a^ \OjX-\- a2y
where the constants are arbitrary, define a non-projective
group of order r+ 4. The group property may be verified
easily. The identical transformation is obtained by taking
a2 = a3 = a5= ... = 0, and ax = <x4 = 1, and the infinitesimal
operators may be written down without much difficulty ; but,
since a general method of obtaining these will soon be in-
vestigated, we shall not now consider these operators.
This group is not similar to the projective group, nor to any
of its sub-groups.
§ 26. In three-dimensional space many of the groups have
long been known ; there is the general projective group of
order 15, viz.
, _ aux + a21 y + a31z + aiX , _ a12x + a22y + a32z + a42
" aux + a2±y + aziz + au' y aux + a2iy + auz + au '
z,_ «13a; + a232/ + g33g + «43
" aux + a2iy + a3iz + au
c a
20 GROUPS OF THE SAME TYPE [26
From this we obtain the linear group of order 1 2 by taking
au = a24 = a34 = 0 ; the linear homogeneous group of order 9
by further taking a41 = ai2 — a43 = 0 ; the special linear homo-
geneous group of order 8 by taking
^11 ' ^21 » ^31
^12 ' ^22 > ^32
tt13 » ^23 > ^33
= 1.
Other sub-groups of the general projective group are : the
group of rotations about a fixed point of order 3 ; the group
of translations, also of order 3 ; and the six-fold group of move-
ments of a rigid body, obtained by combining these two groups
of order 3.
There are very many other sub-groups of the projective
group, but we have now perhaps given a sufficient number of
examples of projective groups in three-dimensional space.
From these groups others could be deduced by transforma-
tions of the variables, but they would not be new types, thus
the groups
x' = aux + a21y + a31z, y' = a12x + a22 y + a32z,
z'= a13x + a23y + a33z,
and , anx 4- a21y + a31 , a12x + a22y + a32
x = » y = >
a13 x + a23 y + aZ3 a13 x + a23y + a33
z' — a13 xz + a23 yz + a33 z2
are of the same type, for the first can be transformed into the
second by the scheme
xx = xz, y1 = yz, zx = z.
§ 27. We may apply the theory of groups to obtain, in terms
of Euler's three angles, the formulae for the transformation
from one set of orthogonal axes to another.
Describe a sphere of unit radius with the origin 0 as centre,
and let the first set of axes intersect this sphere in A, B, G.
By a rotation \}r about the axis OC we obtain the quadrantal
triangle CPQ, and a point whose coordinates referred to the
first set of axes were x, y, z will, when referred to the new set,
have the coordinates xf, y\ z' where
a/= #cos \/f + 2/sin ^, y'= —x sin \j/ + y cos \}/, z'=z.
By a rotation 6 about 0Q we pass to the quadrantal triangle
G1P1Q, and a point with the coordinates x, y, z will now have
the coordinates x'\ y", z", where
x" = x' cos 6 — z' sin 6, y" = y\ z" = x' sin 6 + z cos 6.
27] EULER'S TRANSFORMATION EQUATIONS 21
Finally by a rotation cf> about 0C1 we pass to the axes
0C1, 0A1, 0Bl referred to which the coordinates of x, y, z will
be a*"', /", z"\ where
x'" = x" cos <f> + 2/" sin 0, 3/'" = - x" sin </> + y" cos 0 , z'" = z".
B,
B
A,
If then i2 denotes the operation of replacing x, y, z re-
spectively by
x cos ^ + 2/ sin ty, — X8in\j/ + yco3\}r, z,
S the operation of replacing x, y, z by
x cos 6 — z sin 0, y, xsinO + zcosO,
and T the operation of replacing x, y} z by
x cos $ + y sin <£, — a; sin <£ + £/ cos 0, z,
the coordinates of a point x, y, z, with respect to the first
axes, will be obtained when referred to the new axes 0Alt
0BX, 0C1, by operating on x, y, z with RST, and therefore
22 NUMBER OF TYPES OF GROUPS [27
x'"— (cos 0 cos $ cos \{f — sin (f> sin ^) a;
+ (cos 6 cos <£ sin ^ + sin 4> cos f) y — sin 6 cos <f> . s,
2/"= — (cos 0 sin <f> cos \^ + cos <£ sin \^) a;
+ (cos (f> cos \^ — cos 6 sin <p sin ^) 2/ + sin 0 sin <£ . s,
g"'=Edn 0 cos v/r . a? + sin 0 sin \\r . 2/ + cos 6 . s.
These are Euler's formulae ; if we take
«^, + >/f = €1> 0cos(<J>-^) = «2, flsin(<^->/r) = €3,
and then make e15 ff2, e3 small, we obtain the three infini-
tesimal operators
a a a a a d
J d« dy 3aj dz dy * do;
of this group. These can, however, be more easily obtained
otherwise.
§ 28. An example of a group in three-dimensional space,
which is not derivable from the projective groups by a trans-
formation of coordinates, is
, a1x + b1y + c1 , a2x + b2y + c2
x = 7 > y = 7 ; — »
a3x + b3y + c3 a3x + o3y + c3
r_ {b2cz-b3c2)x + (a2b3-a3b2) (y + xz) + a2c3-a3c2
" (61c3-63C1)aj + (a163-a361) (y-xz) + a1c3-a3c1
If we notice that
,_ ,_,__ {blc2-b2c^x-\-{a1b2-a2b^ (y-xz) + a1c2-a2c1
V ' {blcs-b3c1)x + (a1b3-a3b1)(y-xz) + a1c3-a3c1'
it will not be difficult to verify the group-property.
As the number of variables increases the number of different
types of groups increases rapidly. Thus there are only three
types of groups of the straight line ; there are a considerable
number of types of groups in the plane, but they are now
all known and will be given later on ; in three-dimensional
space there are a very large number of types, most of which
have been enumerated in Lie's works ; but in space of higher
dimensions no attempt has been made to exhaust the types.
CHAPTEE II
ELEMENTARY ILLUSTRATIONS OF THE PRINCIPLE
OF EXTENDED POINT TRANSFORMATIONS
§ 29. Some classes of differential equations have the property
of being unaltered when we transform to certain new variables.
Such transformation schemes obviously generate a group ; for
if S and T are two operations which transform the equation
into itself, or as we shall say operations admitted by the
given equation, TS will also be an operation admitted by
the equation, and therefore S and T must be operations of
a group. This group, however, is not necessarily finite or
continuous.
The differential equation of all straight lines in the plane, viz.
-~ = 0, is an equation of this class ; for from its geometrical
meaning we know that it must be unaltered by any pro-
jective transformation.
Again the differential equation of circles in a plane, viz.
dx^dxz'~'\ ^dx* )dxz'
must admit the group of movements of a lamina in a plane,
and also inversion.
It would be easy to write down many equations which,
from their geometrical interpretation, must obviously admit
known groups ; but more equations exist admitting groups
than we could always obtain by this a priori method ; and
we shall now therefore briefly consider a method by which
the form of those differential expressions may be obtained
which are UDaltered, save for a factor, by the transformations
of a known group. The method will be more fully explained
and illustrated in the chapter on Differential Invariants.
§ 30. In this investigation the underlying principle is that
of the extended 'point transformation.
24 EXTENDED POINT TRANSFORMATION [30
To explain this principle let
x'=x + t£(x,y), y'=y + tri(x,y)
be an infinitesimal transformation ; then
dy' _ dx ^bx tydx)
^<>x <>y ^>x'
~ dx ^<>x ~bydx Ixdx dy^-dx' ' '
since t is a constant so small that its square may be neglected.
If we denote -j- by p, and ~ by p' , and the expression
Ix K^y *x'r <*y*
by it, we have proved that
p'= p + tir.
Similarly we have
dp' dx ^^x ty <ip dx'
~ dx ^x ~&y Zxdx 2>y dx ^pdx'
If we now write r for — this gives, after some easy reduction,
dx °
r'— r + tp,
where
_», , y, »j, .y, yg . yg 8
p~~ da;2"1" V daty dW^ vty8 lxly'p ^yzi
d2/z vc>2/ da?y
The infinitesimal transformation is said to be once extended
when to the transformation scheme
a/=a;+$£, y'=y + tr]
we add p'=_p + <7r;
31] EXAMPLE 25
it is said to be twice extended when we add to these
r'= r + tp,
and so on.
A general rule for extending a point transformation to any
order will be explained in Chapter XX.
We have only considered the extension of an infinitesimal
transformation, but any transformation could be similarly
extended ; the infinitesimal transformations with their exten-
sions are, however, the most important in seeking differential
equations which admit the operations of a known group.
It will be proved in Chapter XX that if we have a group
of transformations, and extend it any number of times, the
resulting set of transformations will belong to a group which
is simply isomorphic with the given group.
§ 31. In order to illustrate the theory of extended point
transformations we shall find the absolute differential in-
variant of the second order ; that is, an expression of the form
f(x, y, p, r), which is unaltered by the transformations of the
group of movements of a rigid lamina in the plane xy.
In this problem the infinitesimal transformation is
af=x+t£, y'=y + tr}, p'=p+tir, r'=r + tp,
where
£=a + cy, i] = b — cx, it = —c (1 +p2), p=-3cpr,
and a, b, c are constants.
SinCG f(x,y,P,r)=f(x + t£, y + trj,p + t7T, r + tp),
and t is so small that its square may be neglected,
(a+cy^+(h-cx^-c(1+^^-3cprh
must annihilate/.
As the constants are independent we infer that
2_, >, «i— y± +(1+^)1 + 3^
Ix 2>y ly u Tix x ^ '^p c *r
must each separately annihilate/.
We conclude therefore that in / neither x nor y can occur
explicitly, so that/ is a function of p and r annihilated by
26 SOME DIFFERENTIAL EQUATIONS [31
it is now at once seen that the required differential invariant
for the group of movements in the plane must be a function
( 1 4- 7)^)-
of —^- 9 that is, of the radius of curvature.
r
§ 32. In the theory of differential invariants we look on the
group as known and deduce its invariants ; a related problem
is : ' given a differential equation or differential expression to
find the infinitesimal transformations which the equation or
the expression admits.'
We know that these transformations must generate a group,
though we do not know that the group will be finite. It
should be noticed, however, that the property of admitting an
infinitesimal transformation at all belongs only to particular
types of differential equations.
Thus if we take the equation
and try whether it admits the infinitesimal transformation
x,= x + t& y/ = y + trj) p' =p + tiri r'=r + tp,
we see that it cannot admit it unless
P = 2xi+2yrj,
for all values of x, y, p, r satisfying the equation r = x2 + y2.
We must therefore have
a2r? , a2 77 a2A ,^-q d2£ . 2 a2f 3
^ + v *x*y *a?)P + \*^~2*&ty)p ~*tfP
for all values of x, y, and p.
Equating the coefficients of the different powers of p to zero,
we get
m *l_0 m In 2 »( ...
(3) 2-^_-^f-3^ + /)=0,
at
32] ADMIT POINT TRANSFORMATIONS 27
From (1) we see that
by differentiating (2) with respect to x, and (3) with respect
to y, and eliminating 77 we get
7>x2^y uMj
that is /'» + 22//(a) = 0,
so that /(a;) vanishes identically.
From (1), (2), and (3) we therefore conclude that
£=<P(x), r) = yf(x) + y\r(x),
and 2f(x) = </>"(«).
From (4) we get
f'(x) + f"(x) + (x* + y2) (f(x)- 2<$>'{x)) = 2aj0(as) + 22/2/(«) + 2^(4
and on equating the coefficients of y2 in this equation we see
that f(x) + 2<t>\x) = 0,
and we conclude that f(x) = <\>"(x) = 0.
By equating the coefficients of y we get \}/(x) = 0 ; while by
equating the terms independent of y on each side we easily
obtain <£ («) = 0, and therefore f(x) = 0.
The equation proposed therefore does not admit any in-
finitesimal transformation.
If we were to treat the equation -^-f = 0 in the same manner,
we should find that the only infinitesimal transformations it
admits are those of the projective group.
Example. Find the form of the infinitesimal transformations
which have the property of transforming any pair of curves,
cutting orthogonally, into another such pair.
Let x'=x + tg, y'=y + tri, p'= p + tir,
be the once extended infinitesimal point transformation ; and
let x, y be the point of intersection of the two curves, and
p and q the tangents of the respective inclinations of the axis
of x to the curves at this point, so that pq + 1 = 0.
We have now to find the form of £ and rj in order that
pq+1 =0 may admit the infinitesimal transformation.
28 TRANSFORMATIONS ADMITTED BY [32
We must have
wherever pq+l = 0. In this and other like examples we
shall employ the suffix 1 to denote partial differentiation with
respect to x, and the suffix 2 to denote partial differentiation
with respect to y.
Substituting - for q in this equation, and equating the
different powers of p to zero, we get
^1 + ^2= °> ii~ri2= °>
so that £ and t] are conjugate functions of x and y.
An infinity of independent infinitesimal transformations
will then have the required property.
§ 33. We know that the differential equation
/k\2 /^x2 /<*M\2
(^)+(^)+(Ji)=0
is unaltered by any transformation of the group of movements
of a rigid body in space ; and we also know that it is unaltered
by inversion with respect to any sphere ; and finally that it
is unaltered by the transformation
x'— kx, y' '= ky, z'= kz,
where k is any constant, that is, by uniform expansion with
respect to the origin. We therefore see that this differential
equation admits a group, and we now proceed to find all
the infinitesimal transformations of this group.
It is a matter of interest to connect this problem with
another one, apparently different, but really the same.
Any curve in space, the tangent to which at each point on
it intersects the absolute circle at infinity, is called a minimum
curve. If x, y, z and x + dx, y + dy, z + dz are two consecutive
points on such a curve,
dx2 + dy2 + dz2= 0.
Through any point P in space an infinity of minimum
curves can be drawn, and the tangents at P to these curves
form a cone ; also through P an infinity of surfaces can be
drawn to satisfy the equation
,^U\2 /SUn2 /<>UX2
and the tangent planes to these also touch a cone ; we shall
now prove that these cones coincide.
testa
lend
with
ntial
lall
with
34] LINES OF ZERO LENGTH 29
On any surface, and through any point on it, two minimum
curves can be drawn ; for in the usual notation we have on
any surface
dx2 + dy2 + dz2 = dx2 + dy2 + (jodx + qdy)2 ;
if therefore we choose dx : dy so that
( 1 +p2) dx2 + 2 pqdxdy + ( 1 + q2) dy2 = 0,
we have two directions for minimum curves through the
point.
Now on any surface, u = constant, which satisfies
/,\ /^u\2 /^u\2 r^u\
we must have 1 +p2 + q2 = 0,
and therefore the minimum lines on the surface drawn through
any point on the surface must coincide ; and, conversely,
surfaces with this property satisfy the differential equation (1).
It follows that any tangent plane, at a given point, to a sur-
face satisfying the equation (l) touches the cone, formed by the
tangents to the minimum curves through the same point ; the
two cones therefore coincide at every point of space, and the
same set of transformations must leave unaltered the two
equations,
(^\2+(^)2+(^)2=0 and dx2+dy2 + dz2 = 0.
This is a particular case of a theorem, to be considered later,
connecting partial differential equations of the first order with
equations of the form
where dxlf dx2, ..., dxn enter the equation homogeneously.
These equations are called Mongian equations.
utive
urves
in be
rial
§ 34. Consider the infinitesimal transformation
af=x + t& y'=y + tr}, z'=z + t£
which has the property of being admitted by the equation
dx2 + dy2 + dz2 = 0.
Since dx'2 + dy'2 + dz'2 — 0, wherever dx2 + dy2 + dz2 = 0,
30 TRANSFORMATIONS ADMITTED BY [34
we say that these two equations are connected ; we now have
the equation
dx (it dx + £2 dy + £3 dz) + dy (^ dx + v2dy + Vs dz)
+ dz((1dx + (2dy + (3dz) = 0
connected with dx2 + dy2 + dz2 = 0.
We must therefore have
(!) £i = r?2 = &> rla + C2=(i+£3 = £z+Vi = 0-
To verify that we obtain these same equations by the con-
dition that the two equations
/Hx2 /^\2 /<*M\2 „ , /<>U\2 /<>U\2 /<>W\2
fo) +M + (s?) - °> and (a) + (sj) + (rJ = °>
are connected, we write down the identities
d d .,t a a . ^
ix = w+t^w + T1iw + Cl^'
^ = ^+H^ + ^ + %iOJ
and, since t is so small that its square may be neglected, we
deduce from these
a a , d 3 a,
^-aaTH^ + ^ + ^J'
i a • 3 3 JK
By the conditions of the problem the expression
must therefore be zero, wherever the expression
/^U-2 /^Us2 /<>U\2
(^)+%)+(sj) ■
is zero, and the equations (1) are thus obtained over again.
35] LINES OF ZERO LENGTH 31
§ 35. We now take
6 = v-z = C3 =ffa y, z\
% + C2 = & + & = ^2 + 77l = 0.
Differentiating t]z + £2 = 0 with respect to y and z, and ex-
pressing the resulting equation in terms of/, we get
Similarly we obtain
a8/ a2f , ay d2f
TT + t4 = °> and -4 + -4 = 0,
oz* oar ^xi ^y2
and conclude that
da2 "" c>2/2 "" t>s2 "
We therefore take
/ = a0 + axx + a2y + a3z + a23yz + a31zx + anxy + al23xyz,
where the coefficients of the powers and products of the
variables are constants, so that
lxiy^z~a23 + anzX> *x ly Iz ~ a%1 + a™y'
By differentiating j/3 + C2 with respect to x, Ci + i3 v^ith
respect to y, and £2 + ^ with respect to z, we have
V23 == ^31 = M2 = " J
and conclude that
^23 = tt31 == ai2 == ^123 == "'
Integrating £x = / = a0 + ax x + a2 y + a3 z
we see that
£ = a0x + %a1x2 + a2xy + a3xz + F(y, z) ;
and since £23 = 0 we see that F(y, z) must be of the form
■^12 (y^ + ^13 (z)> w^ere F12 (y) is some unknown function of y,
and F13 (z) some unknown function of z.
We have now advanced so far that we may take
£ = a0x + ^axa? + a2xy + a3xz + F12 (y) + F13 (z),
v = a0y+ aixy + £ a*y2 + a*yz + F2iix) + F2s(z)^
C= a0z + a1xz + a2yz + %a3z2 + F31 (x) + F32 (y) ;
32 EXAMPLE [35
and from the equations
rii + C> = (1 + £3 = £o + v1 = 0
we next obtain
asV + *»(*) + <*** + Fk(y) = °«
axz + F'zl{x) + a3x + F[z(z) = 0,
a2x + F[2(y) + axy + F^(x) = 0.
We conclude then that
^32 (y) = — i«32/2~ AiV + constant,
F23 (z) = — \ a2z2 + Axz + constant,
with similar expressions for the other functions.
Finally we have
£ = | a^x2 — y2 — z-) + a2yx + a3xz + a0x + a + A2z - A3y,
7] = ±a2(y2-z2-x2) + a3yz + a1xy + a0y + p + A3x-A1z,
£ = *a3(z2 — x2—y2) + a1xz + a2yz + aQz + y + A1y — A2x.
We now have ten infinitesimal transformations admitted by
the equation 3u a du 2 >u 2
and by the Mongian equation
dx2 + dy2 + dz2 = 0.
The ten operators which correspond to these transforma-
tions are
^ d ^ d d d d d d
) —3 ^— ' V^ Z-r-i Z — £Cr— } X- V —
bx dy dz dz dy dx dz dy ° dx
d d d , „ . „ ov d „ d _ ^
x~ - + 2/T" +z^' (y2 + z2-x2)- 2xy- 2zxT->
dx dy dz dx dy dz
(z2 + x2 — y2)- 2xy - 2yz — ,
(x2 + y2—z2)- 2yz 2zx — -
v J 'dz J dy dz
§ 36. Example. Find the most general infinitesimal trans-
formation with the property of transforming any two surfaces j
intersecting orthogonally into another pair of such surfaces.
Let u and v be any two functions satisfying the equation
, , dudV dU dV dU dV
dx dX dy dy dz dZ
then u = constant, and v = constant will be two surfaces
intersecting orthogonally.
36] ON CONFORMAL TRANSFORMATION 33
The equation (1) must therefore admit
We have
M ~ *x ~ "fl te + Vl*y + C] W1
with similar expressions for
^u <)w ^v ^v ^v
}itf' W W itf3 a?;
substituting in (1) and neglecting t2 we see that
lulv lu^v faulty
<ix ~by dy ?)x
is an equation connected with (1).
We are thus again led to the equations
£i = *72=C3> 773 + C2 = ^3 + Ci = £> + *?i = °;
and conclude that the only infinitesimal transformations with
the required property are those found in the last article.
CAMPBELL
CHAPTER III
THE GENERATION OF A GROUP FROM ITS
INFINITESIMAL TRANSFORMATIONS
The identical transformation.
§ 37. From the equations
«4=/< («.»)• (* = i, .-..,»)
which define a group, and from
x'i'=fi(x',b)=fi(x,c),
we have
(!) Ch = <t>k(a>h)> (fc=l,...,r).
Subject to certain limitations on the values of a1} ...,ar,
Cj, ... cr, we can deduce from these equations
(2) &* = **(«> «)i (* = l,..vr).
Now suppose that on taking ax = cv ...,ar = cr the func-
tions ^ (a, c) remain analytic functions of their arguments ;
and suppose further that the values of bl,...,br so obtained
make fi (x{ , . . . , x'n , b1, ..., br) an analytic function of its argu-
ments, within the region over which x\,...,x'n may range;
then as we have always
fi fo e) =fi (x', b) ;
by the hypothesis ah = ck we have
xi=fi(x> «) =/;(«> c),
so that x'i =f{ (x, b), (i = 1, ...,n) :
that is, bk = \jrk (a, a) gives the identical transformation.
38]
THE IDENTICAL TRANSFORMATION
35
Since these values ofb1,...ibr are obtained from the equations
ah = ^kfav •••' ar-> h ■•■a "r/J
it might seem at first as if they would be functions of alt ...,ar:
this, however, is not the case ; they are absolutely independent
of alt ...,ar. To prove this, suppose that
A^. being some functional symbol : then
%j = J i \X^ , . . . , Xn , Aj , . . . , Ar^,
and as Ax, ..., Ar must occur effectively in f> we should have x\
expressed in terms of x'v ...,x'n and arbitrary constants, which
is of course impossible.
§ 38. As an example in finding the parameters which give the
identical transformation we take the case of the linear group
h = n
k = n
We have chi = 2 dhk hi :
putting chi - aM we have
k = n
i ahh hi ~ ahi '
and therefore, since the determinant
a
ii'
a
in
a
nl>
a
nn
cannot be zero, we must have bhi = 0, if h and i are unequal,
and bu = 1.
Of course these values of the parameters for the identical
transformation could have been obtained by inspection of
the equations of the group, but we have preferred to deduce
them by the general method in order to illustrate the theorem
that they are absolute constants.
As we shall very often have to deal with constants such as
bhi, characterized by the property of being zero if h and i are
unequal, and unity if they are equal, it will be convenient
to denote such a constant always by the symbol e^.
D 2
36 ENGEL'S THEOREM [38
Wo should thus express the parameters of the identical
transformation in the general linear group by the equations
,k = 1, ...,ux
but it will not always be necessary to explicitly state the
range of the suffixes.
§ 39. Engel has proved that finite continuous groups do not
necessarily contain the identical transformation.
Thus consider the function due to Poincare'
y
= 22""/,
n =0
which is known (Forsyth, Theory of Functions, § 87, Ex. 3)
to exist only within a circle of radius unity, whose centre is
the origin. It follows that x is an analytic function of y
such that, whatever value y takes, x always lies within
a circle of radius unity. Let x = A (y) : then A is a function
such that, whatever may be the value of its argument, it is
always less than unity.
Take now the transformation schemes x'~ k(a)x. These
clearly generate a group ; for if
x" = A (6) x' then x" = A (a) A (b) x,
and A (a) A (6) = k, k being a constant less than unity, so that
A (a) A (b) = A(c), where
n = oo
c=22-43".
71 = 0
We therefore have the group property, since we can deduce from
xr= X(a)x and x"— \(b)x' the equation #"= \(c)x.
We now have A (b) = -~^,
A (a)
but we cannot take c = a, for that would give A (b) = 1 , which
is impossible, since A(&) is always less than unity.
The method of obtaining the operators of a group.
§40. Let (1) x'^f^a)
be a transformation of the group ; let r— i , expressed in terms
tak
40] THE OPERATORS OF A GROUP 37
of x[, ...,<, av ..., ar be written £^K, ..., afn, a„ ..., ar),
or in abridged notation a^.{ ; and denote by aXk the linear
operator
Let -r: — denote the operation of differentiating totally with
dak
respect to ak any function of x[, ..., x'ni av ..., ar, in which on
account of (1) x[,...,x'n are to be considered implicit functions
of ax, ...,ar.
We have
d , , v ^x[ <^<f) ^n ch£ j^<£
— <p(xv. ..,xn, av...,ar)-^- ^ + "-+^^ + ^
that is, if we express any function of x\, ...,x'n, av ...,ar, in
terms of xv ..., xn , ax ,..., ar by means of the equation system ( 1 ) ,
and then differentiate with respect to ak, we get the same
result as if we had performed the operation
directly on the given function.
If we now keep xv ...,xn, a1,...,ar fixed, x[, . . . , x'n will also
remain fixed; and the increment of any function 0 (#'/,..., x'nr),
where afj = f{ {xf, b) = f$ (x, c),
due to the increment dbk, (the other parameters bv ...,bk_v
bk+1,..., b,r remaining fixed), will be
bX't<j>(x'1',...,xn)dbk.
Since, however, x'{ = fi (x, c) and xv...,xn remain fixed, while
cv ..., cr are functions of av ..., ar, bv ..., br, we may write this
increment in the form
Now <j> (a//, ..., x'n) is an arbitrary function of its arguments;
so that we obtain the identity
Y" — *S •» Y"
t>A*-^Wkc 3
by equating the above two expressions for the increment.
38
THE NUMBER OF OPERATORS
[40
By giving k the values 1, ...,r we have r identities which
hold for all values of x'{ , ...,x„, av ...,ar, bv ..., br, where
ck = <f>k(a>b)> (k=l,...,r).
§41. We now take bv...ibr to be the parameters of the
identical transformation, and since these are absolute constants,
we shall omit the b in bX'/c and write it X'kr simply.
-^- is now a function of av...,ar only, for bv...,br
Tt
are absolute constants ; we write it therefore in the form
akj(av ...,an), or simply a]:j.
Also, since &15 . . . , br are the parameters of the identical trans-
formation, ck = ak , and we have the identities
(1)
^l = aH aXl+'~+alr a^r*
A. r = arl aA j + . . . + arr aA r ,
where the determinant
an>
a
lr
a
rv
. a
rr
cannot vanish identically, that being a condition for the
existence of an identical transformation.
From these identities we deduce
(2)
aX1 = An X1 + ... + Alr X
r>
aXr=\rl Xx + ... + krr Xr,
where A#, ... are functions of av ...,ar; that is, any operator
vjith any implicit set of constants av ..., a is dependent on
Y Y
This theorem is called the first fundamental theorem in
group theory.
§ 42. A group of order r contains exactly r independent
operators.
Lemma. If we have any linear operator of the form
h = r
(1)
2a*
^k
42] IN A GROUP 39
where ak is a function of av ..., ar, we know from the theory
of differential equations that there are exactly (r— 1) functions
of a15 ...,ar which this operator will annihilate. Let Av ..., Ar_x
be any such (r— 1) functionally unconnected functions, then
if/ is any function of av ...,ar, which is annihilated by (1),
we know that it must be a function of Av ..., Ar_x.
It follows that there cannot be any linear operator of the
form (1) which annihilates the n functions fv ...,/„ defining
a group ; for if there were such an operator there could not be
more than (r— 1) effective constants involved in/j, ...,/n, viz.
From this lemma we conclude that there cannot be any
equation system of the form
tC
where A15 ..., \n do not contain xv ...,xn; and therefore there
cannot be any identical relation of the form
k = r
2A& A = o
between the operators aXv ..., aXr when Xv ...,Xr only involve
av ...,ar; that is, the r operators
Y Y
a^ v ••"> a^- r
are independent, and therefore so are the operators
Y Y
If bv...,br are the parameters of the identical transformation,
and b1 + ev ..., br + er an adjacent set of parameters, ev ..., er
being so small that their squares may be neglected, then ex-
panding x'i = fi (xv ..., xns b1 + ev...,br + er)
by Taylor's theorem we have
x'i = xi + 2e& &*> (» = !,...,»);
or since x\ is approximately equal to xi}
k = r
i = n -
Since ^*=2&*j^r'
^
and the operators are independent, we see that there are
40
EXAMPLE
[42
exactly r independent infinitesimal transformations ; and we
see further that the operators of a group, as defined in § 13,
coincide with the operators as defined in this chapter.
§ 43. As an example illustrative of the preceding methods
we take the projective group of space, viz.
(1) tf. =aiixi + a2ix2 + azix3 + au , (i=l,2, 3);
<X14 X^ + #24 #2 ~^~ ^34 ^3 "^" a44
from these equations we obtain (p being < 4)
2>a£
./•
i>
if g < 4,
and
t) ttpg, (Z14 CCj + Ct24 iC2 + tt34 tC3 + tt^
2)aJ^ aiixi Wp + ^2« ^2 ^ ~^" a3i X3 Xp <a4iXp
i*a
Jpi
(<z14 a^ + <z24 x2 + a34 £3 + %)
If j4„„ is the minor of apq in the determinant
M =
a
IV
a41, . . . au
a
u
X;
we have, as the scheme inverse to (1),
Aix x\ + Ai2 x'2 + Ai3x'3 + Au
Au x\ + A^ x'2 + A43 x'z + Au '
Since only the ratios of the constants are involved, we may
take aM as absolutely fixed ; and we get as the operator
corresponding to apq
(2) if-1 {A^ + A^Xz + A^x's + ApJ^ if q< 4.
If q = 4 the operator is
Jf-^^.^ + ^^^ + ^s^ + ^^^+^+^^J-
The identical transformation is obtained by taking apq = €pg :
this gives A pg = e„~ , and the corresponding 1 5 operators are
XV^ Igrs 1,2,8/'
(4)
I XP V ^1 > V + ^ > V + ^3 ,W j
Sa4
^^"■"^da^
(p= 1,2,3).
44] ADDITIONAL EXAMPLES 41
The reader may easily verify that the set of 15 operators
given by (2) and (3) is dependent on the set of 15 given by
(4) ; and also that either of these sets of operators contains 1 5
independent operators.
Examples. Find the infinitesimal operators of
(1) the projective group of the plane ;
(2) the orthogonal linear homogeneous group, viz.
x'= anx + a21y + a31z, y'= a21x + a22y + az2z,
where the constants are such that
x'2 + y'2 + z'2=x2 + y2 + z2;
(3) the linear homogeneous group in n variables ;
(4) the non-projective group given in § 25.
The canonical equations of a group.
§ 44. The parameters bv ...,br which determine the identical
transformation in the group
x2 == Ji \x> a)
give for all values of the parameters av ...,ar
they are therefore the same parameters as those which determine
the identical transformation in the first parameter group (cf.
(3), § 19)-
It also follows from the definition of the functions
akj \av •••' ar)
that the infinitesimal operators of the first parameter group
are Av ...,Ar where
Let now a^, ...,ar° be the initial values of the variables
av ...,ar; let the operator
e1A1 + ... + erAr
be written A ; and the operator obtained by replacing
av ..., ar in A by a^, ..., ar° respectively be written A0.
If X is any linear operator, we shall denote by ex the expression
1 + h.x + hX2+ hJ3+ -• t0 infinity-
42 THE CANONICAL EQUATIONS [44
Wo now take
«k = eotA«k°> (k=l,...,r),
when we have Tta^ = ^oft^'
and therefore, 4>(av ...,ar) being any function of av ...tar,
jt4>(av...,ar) = A0<j>(av...,ar).
We also have -jr AQ = A0 -jt
since the operators are in unconnected sets of variables, viz.
t and a^, ...,ar° ; and therefore
Similarly we have
and therefore the limit of -£ , when t is zero, is
A^{a^ ...,ar°).
Since <f> (av ..., ar) is a function of t and of the initial values
a^, ...,ar°, we have by Taylor's theorem
, /dcf)s t2 /d2(p\
and therefore
t t2
<f>(av...,ar) = (l+J-|^o+^T^o2+--)^(ai°'--"ar0)-
From this formula we deduce
d t t2
^c}>(a1,...,ar) = A0(l + —AQ+ — A2-{-...)4>(a10,...,a1P),
t f2
= Acf)(av ...,ar),
by a second application of the same formula.
A particular case of this second formula is
0) -J^=2e*as&-
44] OF A GROUP 43
The identities of § 41 (expressed in the variables x'x, ...,^4)
[ aX'k = \klX[+... + \krX'r, (Jc = 1, ..., r)
are equivalent to
(2) ai'ki = Xhi ili + •" + Xkr & 5
and therefore, since xx is a function of xx, ..., xn, ax, ...,ar and
thus implicitly of xv .,., xn, ax°, ...,ar°, t, and since
^'i _ t'
dx'- k=ir3==r
Wehave ~dt = 2 kkj£jie8ask
by (1) and (2).
Now the identities (1) and (2) of § 41 are equivalent, so that
k = r
we must have 2 hj ask = *sj I
and therefore
(3) §='!«. &•
We can deduce from the formula (3) a result which will be
useful later ; since
we have the inverse scheme
xt - F{ (x\ a) ;
and therefore, since a^ does not involve t, we see that
d
It follows from (1) and (3) that the operator
that is, the operator ^e r y + ^ )
annihilates every function of xv ...,xn when expressed in terms
of x'x, ...,x'n, ax, ...,ar. If we notice that x'x, ..., a^, ax, ...,
ar, ex, ..., er are all independent of one another, we shall see
that each of the operators X'x + Ax , . . ., X'r + Ar , must have this
property.
44 ALL TRANSFORMATIONS GENERATED [44
If we now take fl^0, . . . , ar° to be the parameters of the identical
transformation, then, when t = 0, x\ = x^ ; and applying
Taylor's theorem we have
^-^+H^;t=0+ 2i^ dt* h=+""
If we write X' for the linear operator
e1X'1 + ...+erX'n
and express any function of x[, ...,x'n in terms of xv ...,xn,t,
ev ..., er we have from (3)
d
-^ 4> {x\, . . . , aQ = X> «,..., .<).
Now .X't/) (x'v ..., a4) is itself a function of a^, ...,a4, so that
d
t-X'$ (x[, ...,<) = X'2^;, ...,<),
and therefore
d2
"ip 9 (*ij • • • j xn) = X <p [xv . . . , a^J,
and more generally
-=— 9 (a^, . . . , xn) = A <p [xv ..., xn).
It follows that the limit of ( —rrz~) is XmXj, and therefore
v dtm /^o
4 = (l + ^X+ ^X2 + ...)a-,. = etxXi.
Similarly we could prove that
y±) (p (xv ...,xn) = e (p \XV ...,xn)y
where X denotes the operator
e1X1+ ...+erXr.
Example. Assuming that
x\ = etxxi:, prove that $ (x[, ..., x'n) = etx$ (xv ...,xn).
Since Av ...,Ar are operators given by
44] FROM INFINITESIMAL ONES 45
where the determinant
all) • • ' air
arv ' ' ' arr
does not vanish identically, these operators are not merely
independent but also unconnected.
A group in n variables with n unconnected operators is
said to be transitive ; if the order of the group is also equal
to n the group is said to be simply transitive.
We now see that the first parameter group is simply
transitive.
Since Av ...,Ar are unconnected operators, and ev ...,er
arbitrary parameters, and av ..., ar are defined by
| ah = et^\ili\ (k=l,...,r),
we know that there can be no functional connexion between
av ..., ar, they may therefore be any parameters whatever.
It follows that if
xi = fi (xv >->XW av • • •' ar)> (* - li — i 7l)>
we can always throw fi (x, a) into the form
eeijri + -+erJrrxi.
When the equations of a group are given in the form
a£ = e^i-^-^2^ (i= 1,...,%),
the group is said to be in canonical form.
Since eeixi + ■■■ + er-^r [a the limit when m =co of
(l + e1X1 + ...+erXr>ln ^
v m '
we see that every finite operation of a group can he generated
by indefinite repetition of an infinitesimal operation.
It should be noticed that the operation of substituting
for x^ ..., xn in any given function of these variables
x[,...,x'n respectively, an operation denoted in the first
chapter of this treatise by Sai, ...,ar, has now been proved
equivalent to operating on xlt ...,xn with ee1x1 + ... + erxrj when
e1, ...,er are functions of a1,...,ar known as the canonical
parameters. We shall sometimes speak of e6'1^'"1^1' as
46 THE METHOD OF OBTAINING THE [44
a finite operator of the group, or simply as an operator, when
there is no risk of confusing it with a linear operator.
When in canonical form, the parameters of a transformation
scheme and its inverse are very simply related.
We have seen that
<p {xv ...,xn) = e (p (x^, ...,xn),
and since this formula holds for any function of x[, ..., x'n we
must also have
e-Ar'<t>(x[, ...taQ = ex c~x <$>{xv ...,xH).
Now just as in elementary algebra we see that
e* e~x '= 1,
and therefore <j>(xv •'•,%„) = e~x <j>(x[, ...,x'n).
A particular case of this general formula is
x{ = e~e^x^~---~erX'r x^,
so that the canonical parameters of any transformation scheme
being ev ...,er, those of the inverse scheme are —ev ..., — er.
Examples, (l) Prove that, X being any linear operator,
xi = e ®ii K1 — 1) ''•)"')
is a group of order unity.
(2) If X and Y are two linear operators whose alternant
is zero, prove that any transformation
**-» — e ^i
is permutable with any transformation
x'a = etYx,
r
§ 45. When we are given the infinitesimal transformations
of a group — and the group is generally discovered through the
infinitesimal transformations — we are given the group in its
canonical form ; the question then arises, How are we to
determine whether a known set of linear operators do, or do
not, generate a finite continuous group 1
This question will be answered in the next chapter, but just
now it will be assumed that XT, ...,Xr are r linear operators,
known to generate a group given by
a^ = eeijri + ...+er xTXh (i = l,...,n).
The group is, however, only given in the form of an infinite
series, involving the evaluation of such terms as
{e1X1+...^erXr)mxii
45] FINITE EQUATIONS OF A GROUP 47
so that we may ask, Can a£, ...,x'n be expressed as finite
functions of xv...,xn?
The differential equation
(e1X1+ ...+erXr)u — 1
has n unconnected integrals ; let these be
(p1 {Xj, ..., xn), ..., <pn {xv ..., xn).
If we take as a new set of variables yv ...,yn where
we see that (e^^ ...+erXr)y1= 1,
and (e1X1+ ...+erXr)yi = 0 if i > 1 ;
and therefore the operator
X = e1X1+ ...+er Xr,
expressed in the new variables, is -r — •
Now we have proved that $ (a^, ..., xn) being any function
of the variables <£ (x\, ..., x'n) = ex4>(xl, ..., xn),
and therefore we conclude that
4n{x^ ..., a£)— $, (x[, ...,<) = <pn (xv...,xn)-4>1(xv...,xn),
while
From these n equations we can therefore deduce the expres-
sions for x[, ..., x'n in terms of xx, ..., xn.
It follows that, when we are given the infinitesimal operators
of a group, we can find the equations of the group in finite
terms if we can find the integrals (f>v ...,<pn of
{e1X1 + . . . + erXr) u = 1 ,
and then solve the equations
«k(a^ ...,x'n) = 4>i(xv...,xn) + l, (i = l,...,n),
so as to express x[, ...,x'n finitely in terms of xv ..., xn.
The functions <f>v ..., (pn will of course involve the arbitrary
parameters ev ..., er .
Example. The operators
h a; — j v— -+0— -j (xy — z) — + yz — byz — >
ly Iz y by bz v u 'lx * by * bz
48 EXAMPLE [45
are known to generate a group ; find the equations of the
group in finite form.
We have to find the integrals of
The subsidiary equations are
dx dy dz _ du
e3(xy-z)~~ et + e2y + e3y2" ex x + e% z + e3 yz " 1
and if we write
Vle.e^—e.f e2 z e2
a = — — » a tan <b = y+ ~ > a tan 0 = — I - >
2e3 ° 2e3 x 2e3
these equations become
cZlog# 7 , 7„ \/4e1e,-e„2 7
= d<p = d6 = L? 2- du.
tan <f) — tan 9 2
So that
u = — = =. tan T iiJ f==
w 4exe3 — e22 w4exe3—e22 v/4e1e3— e22
CC COS (b
is an integral of the proposed equation ; and —■ , and <p — 0,
are functions annihilated by the operator cos
e3 (ocy-z) — + {ex + e2y + e3 y2) — + {exx + e,z + e3yz) — ;
that is e^ + ^ZX + e^ and *-** , !
are annihilated by this operator.
The finite equations therefore of the required group are
e3z'2 + e2z/x' + e1x'2 _e3z2 + e2zx + e1x2
ezy'2 + e2y' + el e3y2+e2y + e1
z' —x'y' z — xy
2 e3 y'zf + e2 {x'y' + z') + 2e1x' ' ' 2e3yz + e2 (xy + z) + 2e1x
V4c1e3-e2a V4e1e3-e22
= tan"1 3* 2 + 1 ;
V4e1e3-e22 V*e1e3-e2
46] THE INFINITESIMAL OPERATORS 49
and if we were to solve these, and thus express x', y\ z' in
terms of x, y, z, we should have the finite equations of the
group in canonical form.
§ 46. There is generally considerable difficulty in expressing
the equations of a group in finite form when we are given the
infinitesimal operators; but for most parts of the theory of
groups the knowledge of the forms of the infinitesimal opera-
tors is of more interest than the knowledge of the finite form ;
and the most important result which we have proved in this
chapter is that every transformation of a group may be
obtained by indefinite repetition of a properly chosen infini-
tesimal transformation.
Thus if we take the binary quantic
u — a0xP+pa1xp-1y + ...,
and apply the linear transformation
af=l1x + m1y, y'=l2x + m2y,
we get u = a'Qx'p +pa[x/p~1 y' + ....
From the identity of these two expressions for u, we deduce
(1) a'0=a0lp + pax I*-1 l2 + ...,
a[ = a0lp~1m1+ ...,
with similar expressions for a'2, ...; and the problem of the
invariant theory is the deduction of the functions which have
the property
f(a'0, a[, ...) = Mf(a0, a1} ...),
where M is a function of lx, m19 l2, m2 only.
Now the equations (1) are easily proved to be the finite
equations of a group of order four; but they are of little
use in the invariant theory in comparison with their four
infinitesimal operators
3 o a o a
0 doj 1 la2 * da3 '
d 3 d
* lap_x P 1 lap_2 P'2 lap_3
d d d
oa0 l<ia1 -oa2
3 n d 3
a,- H2a2T r3a3- — + ... .
13a1 Ala2 ^az
CAMPBELL
E
50 THE INFINITESIMAL OPERATORS [46
A like result holds for most of the applications of continuous
groups ; thus, one of th'e questions to which the theory is
applied is the investigation of those linear partial differential
equations, which are unaltered by the transformations of a
known group ; we know that every equation, which admits
all the infinitesimal transformations, will admit all the finite
transformations of the group, for the latter can be thrown
into canonical form ; and it is much simpler to find the forms
of differential equations admitting known infinitesimal trans-
formations than the form of those admitting known finite
transformations.
CHAPTER IV
THE CONDITIONS THAT A GIVEN SET OF LINEAR
OPERATORS MAY GENERATE A GROUP
§ 47. We have proved in the last chapter that a group of
order r has exactly r independent linear operators, in terms
of which all other linear operators of the group can be ex-
pressed ; and when these operators are known the group is
also known in canonical form.
If Xx, ...,Xr are any r independent operators of the group,
we can express all other operators of the group in terms of
these ; there is therefore no unique system of operators ; thus,
in the group of rotations about the origin,
X = y- 2-1 Y—z- #— > Z=Xr y —
cz i>y ox oz oy ox
will be three independent operators ; but so also would be
axX + bxY+cxZ, a2X + b2Y+c2Z, a^X + ^Y+c^Z,
provided that the determinant
a,,
K
<H.
a2,
K
c2
<*3>
K
C3
did not vanish.
We shall, however, suppose that we have fixed on some one
set of independent operators, in terms of which the others
are to be expressed.
The proposition which, with its converse, will form the
subject of the present chapter may now be stated.
If XXi ..., Xr is a set of independent operators of the group,
the alternant of any two of these is dependent on the set ;
that is . .
XiXj—XjXj — (Xt, Xj) =2<cijliXhi \j _ i} ..#j r/
where the symbols c^ , ... denote a set of constants, called the
structure constants of the group ; these constants are fixed,
E 2
52 SECOND FUNDAMENTAL THEOREM [47
when once the set J,, ...,Xr is fixed, but they vary with our
choice of the set.
The converse of this theorem is, if Xlt . . . s Xr are any r
independent linear operators such that
(Xi> Xj) = 2 cijk Xh J
then Xx, ...,Xr will be the operators of a group, which will be
finite and continuous, and will contain the identical trans-
formation ; the canonical form of the group will be
We have proved that in operating on any function of
aflt ...,x'n, ax, ...,ar, where we regard x1, ...,xn as fixed, and
a?i,...o4 as varying, through being implicitly functions of
xx,. ..,#„, ax, ..., ar, we have (§ 40)
dak ° * dafc
Since then ^ = — = -= — = — 5
aakaah aakaak
we have
U + 4) (0xi + jL) = („xi + A) (aXi + 2.) :
expanding this we get
"^aX'h + aX'^ + ^-aX',,
- aX'h aX'k + «x'h ^rk + ^rh aX'k '
This identity is true for all values of o^,...,ar, x'x,...,x'n\
we may therefore replace x\ by x.t, and in the notation of
alternants we have
M GA. A) + (^ ' A) + («*&> ^) = o-
From the set of identities obtained in §41, viz.
aXk = \klXx+...+ Xkr Xr
in which Xki , ... only involve ax , . . . , ar , we have
/ ^ y N _ ^hl y . , <%r v-
I ^ ' aA^ h I = "n -A 1 + . . . + -r -A r ,
v*)^ a ns Zak L <>ak r
( y * \ _ *A&1 Y- ^Atr xr
I a A A ' ^ J = =\ -A i — . . . r A ,
48] EXAMPLE 53
and therefore conclude from (1) that
(aXk ' a^h) = Aftfcl ^l + ■ ■ ■ + XJchr %r >
where the functions A^, ... only involve ax, ...,ar.
This identity holds for all values of the parameters %,..., ar ;
we therefore take a15 ...,ar to be the parameters of the identical
transformation, and the functions A.fcW, ... now become absolute
constants and give the identities
k = r
(2) (**,*;) =2 *«/***'
This is called the second fundamental theorem in group
theory.
Example. The equations (1) of § 46 are those of a group
of order four, with the independent operators Xlf X2, X3, X4,
where v _ ^ , n 2> , , ^
1 "^ l<)a2 ^ * ^Op
X4 = a, l-2a2- — + ...+pav
4 x cKXj 27ia2 1 p^ap
and we may verify that
(XliX2) = pX3-2X„ (XXiX3)=0, (X^XJee^,
(X2, X3) = — X2, (X2, X4) = — X2, (X3, X4) = 0.
If we take as the four independent operators of the group,
Ylt F2, F3, F4, where
Fj = XXi F2 = X2, F3=£>X3— 2X4, F4 = X3,
we see that the group has the structure
{YX,Y2)=Y3, (F2, F3) = 2F2, (F3, F1) = 2F1,
(Flf F4) = 0, (F2, F4) = 0, (F3, F4) = 0.
§ 48. We now know that unless a system of linear operators
is such that the alternant of any two of them is dependent on
the set, they cannot generate a finite continuous group ; but
more important, and, at the same time, more difficult to prove,
is the converse theorem, viz. that any operators which satisfy
these conditions will generate a group.
54 FORMAL LAWS OF COMBINATION [48
Before proceeding to prove this we shall consider some
formal laws according to which the symbols of linear operators
are combined.
Let y and x denote two linear operators, and let yx denote
yx—xy, y2 denote yxx — xyx, ys denote y2x—xy2, and so on.
The identity
n « i n(n— 1) „ „
xny = yxn — nylxn~1-\ -y2x n~2 — ...
may easily be proved by induction ; for it is obviously true
when n = 1 ; assume that it holds for all values up to n, then
-j.i n „ i n(n—l) „
xn+1y = xyxn — nxy1xn~l-\ J — - — -xy2xn~2 — ...,
and as xyr_x = yr-\x~Vv> we have
„,, „,, _ n(n— 1)
xn+1y = yxn+1 — nyxxn-\ — — -y2xn ~x — ...,
^ !
—yxxn + ny^af1'1 — . . . ,
so that the identity holds universally.
If we denote by [y, xr] the expression
yxr + xyxr~l + x2 yxr~2 + . . . + xr y,
we next prove the identity
[v xr i = yxr y^'1 4. a* xr~2 ■ < iy yr
L^'(r+l)!J l!r! 2!(r-l)! 3!(r-2)! " v '(r+l)l
Assuming that this identity holds for all values of r up to
n — 1 , then
r Xn -> r *n"S SBn,V
_ ya" ^g;"-1 , / ..n-iVn-iX x"y
' ll(n-l)l 2l(n-2)l'r'"^y > n\ "%! '
Now we have proved that
xny = yxn-ny1xn~1+ — K—~ — ;-y2xn~2— ...,
so that by addition of similar terms in the two series we get
r x*i / „/Fn y,xn~x (-l)nV«\
48] OF LINEAR OPERATORS 55
and as the identity holds when n = 1 we conclude that it
holds universally.
We have of course similarly
- _ v, xT y8+ixr-
b__±__ I . _ y±±_ i/s+i^
8i (r+l)!J"' l!r! 2\(r-l)r""
Examples.
(1) Prove similarly by induction the formula
„ i i* fa — 1 ) « o
yxn = xny + nxn-lyxA — — -xn~2y2 + ...,
d d 5
(2) If V = %^ +2aita~ +'~+PaP-i)
y
<>aQ 1<>a1 r P~ldap
prove that
2/ (2/i + 2) = 2/i 2/ ; 2/2= -2 «, 2/3 = 0, 2/4= °> — •
(3) Prove that y and as being as defined in example (2),
yxr = xr y + rxr~x {yx — r + 1),
2/i 2/r = 2/r(^i + 2r).
*(4) Apply induction to deduce from (3) the more general
formula
rr-l „.8-l
+
s!r! 'r!s! (r-l)l(s-l)!^1 ^'
^-2 ^1-2 (yt — r + s) (y1 — r + s— 1)
(r-2)!(s-2)! 1
af"3
2/g~3 (^-r + s) (^-r + s-1) (^-r + s-2)
+ (r-3)!(s-3)! 1 2 3 "l"""
(5) Prove that # and y being any linear operators,
yx2 — 2xyx + x2y
is a linear operator.
* A generalization of the formula of Hilbert, see Elliott, Algebra of Quantics,
p. 154, Ex. 5.
56 FORMAL LAWS OF COMBINATION [48
(6) Prove that
yr = yxr — rxyxr~l H — —— — - x2yxr~2 — ... + ( — 1 )r xry.
§ 49. Let
- — - = 1— «ji + a2t2— a3£3 + a424 — a5t5 + a6t&— ... ;
then, if Bx, Bz, ... are Bernouilli's numbers,
«2« = (-1)B"17ffi| ' alld tt3 = «5 = «7 =••'= 0.
We shall now prove the identity
If we substitute for each expression in brackets the series
to which we have proved it equal, we find that the coefficient
of yxr on the right is — > and that the coefficient of ysxr~8 is
r_n«r~ - ai . -l. <h \
v ' ^(s+l)!(r-s)I s!(r-s)! (s-l)!(r-s)! "V
By equating the coefficients of the powers of t in
t = (et-l)(l-a1t + a2t2-a3t3 + aiti~...)
we see that the expression in brackets is zero, and therefore
the identity required is proved.
Example. If
z = y + a1y1 + a2y2 + a3y3 + a^ +...
and zr = zr_1x — xzr_1, (r = 1, 2, 3, ...),
prove that _ z± z^ zB
2/_0~2! + 3! 41 +"-
We now let
z — y + a1y1 + a.2y2+ ... to infinity,
then, from what we have proved, we have
y = y,
yx= Ifaxl+aM,
*Aj r* Us -i p U/ •
• ■•••••
50] OF LINEAR OPERATORS 57
Adding these expressions we get
(1) yex = z + [z, — ] + [z, — ] + ... to infinity.
Now if t is a constant so small that its square may be
neglected, {x + ^)r = ^ + f ^ ^ .
and therefore from (1), if we neglect t2,
(1 +ty) ex = 1 + x + tz + —Ax + tzf + —(x + tz)3+...
We can now say that, if t is a constant no longer small,
(l+ty)ex = ex + t3 + t2R,
where R is sortie operator formed by combinations of the
symbols x and y.
§ 50. We now suppose that
%Xj zzz. €>2 -A. j "T a • • "t" 6~ -^x ~ ,
2/= ^Xj+^. + ^Z,.,
where e1,...,er and e15...,er are two sets of parameters, and
X15 ..., Xr linear operators such that
From these conditions it follows that, if 0 is the linear
operator deduced from x and y by the law
z = y + a1y1 + a2y2+...,
then 0 is equal to
Cj ui j T- • • . T Cj. u\. y. ,
where c1} ...,cr are a set of constants, which are functions of
Bp ..., er, ej, ...,er, and of the absolute constants cty&, ....
From the definition of z we see that these constants
c15...,cr are analytic functions of et, ...,er, ej, ..., er; and
therefore the coefficients of the differential operators in z
will be finite, provided that e1} ..., er, els ..., er do not exceed
certain fixed limits. It now follows that, ex and ex + ts being
two operators whose effects on the subject of their operations
are not in general infinite, the effect of R on any such subject
cannot be infinite.
If we now denote by xx the operator x-\ , where m is
some integer, then xx will be a linear operator dependent on
58 PROOF OF CONVERSE OF [50
X, Xr\ and the result at which we have arrived may be
thus expressed
(1) (l +!.)<? = &+1-R.
Similarly we must have
(2) (i + X)^=6?*+_Li^J
rnx/ m~
where xx has replaced x in (l).
So we have
(3) (l + y~)ex> = e™ + —r,R.y,
in,' nx
(l + -^-)eXm-1= eXm+ —„Rr
Multiplying (1) by (l + %) , (2) by (l + ^) , and
so on, and then adding we obtain
(i + y.)"^=^+±Mi + ^)Mij+ (i + i-)m-\+...).
v nx} m2\v nxJ v mx' l /
Now let m become infinite; from what we have proved
for R we see that
A((l + -^) R+(1 + 1L) R1+...)
m-\v nxJ ^ m' /
is an operator whose effect on any subject on which it
operates is zero when rtx is infinite ; and because xm is always
a linear operator dependent on Xx, ...,Xr whatever m may
y m
be, and because also the limit of ( 1 + — ) is ev we conclude
^ nxJ
that e"ex = e*
where X is some linear operator dependent on Xx, ...,Xr.
§51. We can now easily prove that a set of operators
which have the property
fc«=r
(1) (*«,*}) =2 <tyb*»
will generate a group.
From the definition of a group in canonical form, we see
that what we have to prove is, that if
X = Xj A x + . . . + \r Xr ,
Y = fx1X1+...+nrXr,
52] SECOND FUNDAMENTAL THEOREM 59
where \v ..., Aw and p^, ..., pr are two sets of parameters, and
if Y' denotes the operator obtained by replacing xi in Y by
% where x\ = ^x{, (i=l,...,n),
then eT x\ = rfV*i+...+*rJ&fl^
where vx, ...,vr are a set of parameters, which are functions of
Xj, ..., Xr, fij, ...3ftri and of the structure constants c{jk, ... .
Now eY'x'i is afunction of x[, ..., x'n, and therefore by § 44, (4)
and as we have proved that
we now conclude that the conditions (1) are sufficient as well
as necessary in order that Xti ...,Xr may generate a group.
§ 52. To find i>,, ..., i>r in terms of A15 ..., Ar and /Xj, ..., /xr
would be to find for the group in canonical form the functions
fo (*!,..., A,* ix1,...,nr), (fc=l,...,r),
which define the parameter groups.
Without attempting to perform the calculations necessary to
find these functions, we can see the terms of the first degree in
the expansions of pv ...,vr respectively, in powers of A15 ..., Xr,
bu, ...,/xr ; for, neglecting all products of these parameters, we
have
^xx + ... + \rxr enlx1 + ...+frxr
= (1 + \1X1+...+\rXr) (1 +im1X1+ ... +\J.rXr),
= l+(A1 + M1)^1+... + (Ar + Mr)^r5
and therefore vJc = X]i + n](+ ... ,
where the terms not written down are of higher degree than
those which are written down.
It follows that any operation of the group
x\ = e^1 + • • ■ + erXr xi
can in general be written in the form
xfi = e*^1 etiX* . . . etrXr xt.
To prove this we recollect that the necessary and sufficient
conditions, that r functions of r variables should be capable of
assuming r assigned values, are that the functions should be
unconnected. Now we have proved that g^e^a ... grXr js
equal to e"i-*i + "«^* + • • • + v* xry
where vk = tk + ... , (k = 1, . . ., r) ;
60 EXAMPLE [52
and as L, ...,tr are unconnected so must vls ..., vr be uncon-
nected: by a suitable choice of the parameters tlt ... they can
therefore be made to assume the respective values e1? ..., er.
§ 53. Example. Prove that the operators
a a _ a a _ a a
X=Vr Zxr-s Y=Z- X xr~ > ^ = 3 r £/ V >
^ 2)0 c>2/ a# as 02/ ox
generate a group.
We have
(7,Z) = -X, (Z,X) = -7, (X,Y)=-Z,
and therefore by the converse of the second fundamental
theorem these operators generate a group.
If now we require the equations of this particular group
in finite form, we may proceed as follows.
The most general operation of the group is
Let x'— etlXx, y'— etlZy, 0'= e^xz,
so that
= 2/ cosij— ^sinij.
Similarly we see that
z'= y sin tY + z cos tx
and #'= a?.
We now have
^y^xx _ ettY' x'= x' cost2 + z* sint2= x",
et2Yet1xz _ gfcs-Z'/ _ s/cos^ — £c'sin^2= is".
And finally we get
x'"= x" cos tz—y" sin i3,
2/"'= x" sin £3 + j/'7 cos t
:]■
From which equations we could express x"\ y'" , z'" in terms
of x, y, z, and the parameters tx, t2, t3
"3*
54] EXAMPLE 61
§ 54 If m of the operators of a given group Xv ..., Xr are
such that the alternant of any two of them is dependent on
the m operators, then these m operators will themselves form
a group, which will of course be a sub-group of Xx, ..., Xr.
Example. Find the projective transformations which do
not alter the equation
x2+y2 + z2 = 1.
The most general operator of the projective group is
Zx
(a0 + axx + a2y + a3z + x (exx + e,y + e3z)) —
+ (b0 + bxx + b2y + b3z + y (exx + e2y + e3z)) ^
+ (c0 + cxx + c2y + c3z + z (exx + e2y + e3z)) ^ ;
we must therefore have
x (a0 + axx + a2y + a3z) + y(b0 + bxx + b2y + b3z)
+ z (c0 + cxx + cxy + c3z) + (exx + e2y + e3z) (x2 + y2 + z2) = 0
for all values of the variables such that
x2 + y2 + z2 = 1.
This gives ax = b2 = c3 = 0,
a2 + bx — a3 + cx = b3 + c2 = a0 + ex = b0 + e2 = cQ + e3 = 0,
so that there are six operators admitted by the given equa-
tion, viz.
tr / q -\ ^ ** ^ T7" ^ ^
X, = (X2— 1) — + #ty — + %Z—, F, =-?/- 0 r— >
1 v '^x^^y ^z x u Iz 2>y
A„ = wc^— + (v2 — l)r- + yz^—> Y2 — Zt x—,
2 J Ix vy 'ly y *z 2 c>x Zz
XT" ^ ^ . _ _. 5J TT ^ ^
3 da ^<>y K }^z 3 <)y *<sx
We find that
(X2, X3) = F1} (X3, Zx) = F2, (Xx, X2) = Y3,
(Y2,Y3)=-7X, (Xx>Yx) = 0,
(¥$, Yx) = — Y2, (Fj, F2) = — F,, (X1 , F2) = — X3,
(X15F3) = X2, (X2,F2) = 0,
(X2, Fj) = X3, (X2, F3) = — Xx, (X3, Yx) = —X2,
(X3,F2) = X15 (X3,F3) = 0;
62 TWO RECIPROCAL SUB-GROUPS [54
these six operators will therefore generate a group, and of
this group Tlt F2, F3 will form a sub-group.
We could of course have foreseen that such operators must
generate a group, from the general principle that if T1 and T2
are any two operators admitted by an equation, then Tx T2 is
also admitted ; and therefore the alternant T1 T2 — T2 Tv which
is a linear operator, is also admitted ; and must therefore be
connected with the operators which belong to the group
admitted by the equation.
Also in this example the group must be a finite one ; for, if
it is a group at all, it is a sub-group of the general projective
group.
§ 55. If Xj, ..., Xr are the operators of a simply transitive
group, and Yv ..., Ys the operators of a second such group,
and if the alternant of (Xi , YA is zero for all values of i and j,
then it is clear, from the canonical forms of the groups, that
any operation of the one group is permutable with any opera-
tion of the other group ; such groups are said to be reciprocal.
In the group we have just considered, taking as our set of
six independent operators
Zx =Xx + iYXi Z2 = X2 + iY2, Z3 = X3 + iY3,
W^X.-iY,, W2=X2-iY2, W3 = X3-iY3,
where i is a square root of negative unity, the group has,
with respect to these operators, the structure
(Z2,Z3)=-2iZ1, (Z3,Z1) = -2iZ2, (Z1,Z2)=-2iZ3,
(W2,W3) = 2iW1} (W3,W1) = 2iW2, (W1,W2) = 2iW3,
It is easily proved that each of the sub-groups Zx, Z2, Z3
and W±, W2, W3 is simply transitive; they are therefore reci-
procal sub-groups.
§ 56. Examples.
(1) If u, v, w are three quadratic functions of x, prove that
s a d
U — > V— j 1U —
ox ox ox
generate a group.
(2) Prove that — and #3 —
ox dx
cannot be operators of a finite continuous group.
57] EXAMPLES 63
(3) Find the relations between the constants a, b, c, d
in order that
(ax + by) - — \-(cx + dy) — and x z—
v J/ 7ix v J/ Zy Zy
may be operators of a group of order three.
(4) Prove that
?/- 0— - and (xi—,ui—zi)- — v2xy — + 2xz —
J }>z ^y v ' Saj * dy ^
are the operators of a group of order two ; find the finite
equations of the group, and hence verify that the group is
an Abelian one.
(5) Prove that
y- 3— and (y2—z2—x2)T- + 2yx— + 2yz — -
v Zz 2>y %y * Zx . 7>z
are two operators of a group ; and find the other operators of
the group of lowest order containing these two.
§ 57. Example. Prove that a finite group containing
3 d ~b d
Xx—, xz—, y^—> 2/^— >
2>£c' *y v Ix * oy
cannot contain an operator of the form Uz — H v z— where u
c ex cy
and v are homogeneous integral functions of x and y, of degree
higher than unity.
The principle which enables us to prove this theorem is
that a group which contains two operators must contain their
alternant. The alternant of two operators which are both
homogeneous is then itself a homogeneous operator of the
group ; and if the degrees of the two operators are r and s
the degree of the alternant is (r + s— 1). If then the group
is to be finite, there must be a limit to the degree in which
x and y can be involved in an operator; we may therefore
suppose that there is no operator of degree higher than that
of the operator
Uz- +V~'
dx oy
Now suppose that Uz — f- v r— is of degree r, and can exist
rc dx cy
64 EXAMPLE [57
in a group which contains
3 3 d d
dx oy u dx oy
As we cannot have u and v both identically zero, we may
suppose that u is not identically zero.
Form the alternant of u r — Y Vi-~ with x ^— > and we have
an operator ux ^ — I- v1 — also of degree r ; in ult however, y is
of lower degree than it is in u.
By forming the alternant of ux — + v1 ^- with a? — > and
proceeding similarly with the resultant operator, we see that
the group must contain the operator
when i; is some homogeneous function of x and 2/ of degree r.
Denote this operator by Y, and x — by X, and let
Y,= YX-XY, Y^Y.X-XY^... \
then Fr+1 = (r-ir+1^^3 since *r+1^Fi = 0.
Now r > 1 : so that the group, if it exists, must contain the
operator xr r— •
Forming the alternant of xr — and y — > we see that the
group will contain yxr~l7—> and therefore
that is, yx2r~2~ •
But, since r > 1 , this operator is of degree higher than r, and
therefore we may conclude that the proposed group cannot
exist.
59] STRUCTURE OF PARAMETER GROUP 65
§ 58. We proved in § 44 that Ax, ...,Ar, the operators of
the first parameter group, were unconnected ; and that
Xv ...,Xr being the operators of the group of which Ax, ...,Ar
is the parameter group
Xx + Ax, . . ., X r -+- Ar
each annihilated any function of xx , . . . , xn , when expressed
in terms of x[, ...,cc'n and av ...,ar.
It follows that the alternant
(X'i + A^X'j + Aj)
annihilates such a function ; and therefore so also does
k = r
(X'4 + Ait X'j + Aj) - 2 cijh {X'k + Ah).
Expanding the alternant and noting that
k=r
C^t#» Xj) — 2* cijk X'k
vanishes identically, we conclude that
!c = r
(Ait Aj) — 2,cijhAh
annihilates any function of xx , . . . , xn , when expressed in terms
UJ. %h-t j • • • j *b.n J ttl j •••) Ct',« •
Now this operator does not contain x[,...,x'n, and there-
fore, from what we proved in § 42, it cannot annihilate the
■ functions which express xl,...,xn, respectively, in terms of
xfx, ...,x'n, a}, ...,ar, unless it vanishes identically; we must
therefore conclude that
h-r
{Ai,Aj) =2^cijhAk;
that is, the first parameter group has the same structure con-
stants as the group Xx , . . . , Xr .
§ 59. The theorem of § 41, known as the first fundamental
theorem, tells us that if
(!) a/=/i(3i,...,an, a1,...,ar), (i=l,...,n)
are the equations of a group, and
the operators derived from (1), by the method explained in
§ 40, then
CAMPBELL
66 FIRST FUNDAMENTAL THEOREM [59
(2) aXk = \klXl + ...+\krXr, (k=l,...,r),
where A^.-, ... are functions of ax, ..., ar, and
Y Y
are the operators obtained from
Y Y
er1- is • • • s a r
by substituting therein, for a15 ...,ar, the parameters of the
identical transformation.
The converse of this theorem can now be proved.
Let (1) denote a system of equations known to involve the
identical transformation ; we can form the operators
aX1,...,aXr and Xv ...,Xr
from the equations (1) without presupposing any group pro-
perty of those equations ; the converse theorem then is, ' if the
equations (2) are satisfied, then the equations (1) will define
a finite continuous group.'
On referring back to § 44, it will be seen that the two facts,
firstly that (1) involved the identical transformation, and
secondly that its operators were connected by the equations (2),
involved as a consequence that
If therefore we can prove that the alternants obtained from
Xl,...,Xr are dependent on Xv ...,Xr, then the converse of
the second fundamental theorem will show us that the equa-
tious (1) are the equations of a group.
Now the equations of § 40, viz.
aah dak
are independent of any group property in the equations (1) ;
and (3) and (2) were the only equations used in § 47 to deduce
(2) of that article. We conclude therefore that the facts, that
x^= j^(x^, ...,xn, ttj, ...jCfr^, \i = i,...,n)
involves the identical transformation, and that its operators
are connected by the equations (2), are sufficient to ensure
that the equations (1) are the equations of a group.
This is converse of the first fundamental theorem.
CHAPTER V
THE STRUCTURE CONSTANTS OF A GROUP
§ 60. If Xv X2, X3 are any three linear operators whatever
we have from the definition of an alternant
(1) (Xl,X2) + (X2,X1) = 0.
Also from the same definition
(Xls (Z2, Z3)) = Xx (X2, Z3)-(Z2, X3) X,
— X\X2XZ — X1XZX2 — -^2 ^3 ^1 + ^3^2-^1
and therefore
(2) (Xv (X2 , X3)) + (X2 , (X3 , X,)) + (X3 , (Xl3 X2)) = 0.
This equation will be referred to as Jacobi's identity.
If Xv ...,Xr are r independent operators the second funda-
mental theorem has shown us that
k = r
(3) (XiiXj) =^2CijkXk>
if, and only if, these operators generate a group.
From (1) we then have
k = r
^(Cijk+Cjik)xk = °;
and therefore, since the operators are independent,
cijk + cjik = °-
Again by (3) (Xj, (X{, Xh)) is equal to
h=r h=r h=m=r
(Xj> ^ cikh Xh) = 2 cikh (Xj> Xh) = ^ cikh cjhm Xm »
so that, applying Jacobi's identity, we have
h — m = r
2* (cikh cjhm + ckjh cihm + cjih ckhm) Xm = °-
F 2
68 THE THIRD FUNDAMENTAL THEOREM [60
Since the operators are independent we must therefore have
h = r
2/ (cikh cjhm + cltjh cihm + cjih ckhm) ~ °*
The constants then which occur in the identities
h = r
{x^ Xh) ~ 2 cikh Xh
are such that they satisfy the system of equations
(cikj + ckij = °>
(4) J»-r
( 2* (cikh cjhm + chjh cihm + cjih ckhm) ~ °>
where i, k, j, m may have any integral values from 1 to r.
These constants are the structure constants of the group
corresponding to the operators Xv ..., Xr.
The third fundamental theorem in the theory of finite
continuous groups is that the structure constants of any
group must satisfy these conditions ; and the converse pro-
position is that any set of constants, satisfying these conditions,
will be structure constants of some finite continuous group.
A set of constants satisfying the conditions (4) is called a set
of structure constants of order r ; what we are now about to
show is, how, when we are given any such set of structure
constants, r unconnected operators Xlt ...,Xr, in r variables,
can be found such that
(X%> Xj) — 2 cijh %k ;
that is, we shall find r operators generating a simply transitive
group, with the given constants as its structure constants.
Groups of order r with the given set of structure constants
may exist in a number of variables greater or less than r ;
and the method of obtaining types of such groups will be
investigated in Chapter XI ; in this chapter, however, as we
are only concerned to prove the converse of the third funda-
mental theorem, it will be sufficient to prove the existence of
a simply transitive group with the required structure.
k = r
§ 61. If xt =2aJM«&. (* = l r)
is any linear transformation scheme, whose determinant
7*1' • • • ^'7*7*
61] AND ITS CONVERSE 69
does not vanish, and x\ = 2 Au xk
is the inverse scheme, then, cikh, . . . being any other set of r3
variables, and c'ikh, ... another set connected with the first set
by the equation system
h = r p = q = r
i1) 2 ahs Cikh = 2 aip akq Cpqs >
we see that, since the above determinant does not vanish,
(1) must give c'ihh , ... in terms of cikh ,....
From the fact that in the notation of § 38
p = r
2* Apt akp = eft ,
we easily verify that
h = r
p = q = r
2 -A-hs cikh — 2* -A-ip -a-fiq CpgS ',
and therefore cikh, ... are given in terms of c'ikh, ....
It will now be proved that if one set ciJch, ... satisfy the
system of equations (4) of § 60, so will the other cihh, ....
To prove this, multiply (1) by citmcsm-, and sum for all
values of h, s, m, p, q, when we shall have
h = s = m = r m = p = s = q = r
^ ahs atm cikh csmj — 2 aip akq atm cpqs csmj •
Since by (1) the left hand member of this equation may be
to = h = r
written ^ n / j
^— amj cikh Htm
we see that
m = h = r
^ amj (cikh chtm + ckth chim + ctih c'hkm)
is the sum of a number of terms which vanish by the con-
ditions (4) of § 60.
We therefore conclude, since the determinant does not
vanish, that
h = r
2 (cikh cMm + ckth Him + ctih Hkm) = °
for all values of i, k, m, t.
To prove that cikt + c'm = 0,
70
THE NORMAL STRUCTURE
[61
interchange i, k, in (1) ; we then get
h = r p=q=r
2* ahs ckih = 2* aiq ahp Cpqs'
Adding this equation to (1), from the conditions (4) of § 60
we must have j , j _ n
ciM + Ghit — u*
Suppose now that we have a group with the structure con-
stants cikh, ..., the corresponding operators being Xlt ...,Xr.
If we take as a new set of operators Yx, ..., Yr where
k=r
(2) Yi = H"ikXk>
then it can be at once verified that c'ikh, ... are the structure
constants of the group corresponding to F1? ..., Yr. The con-
clusion we draw is that when we can find a group with the
structure constants cikh,... this group has also the structure
constants cikk, ... corresponding to another set of independent
operators.
We often take advantage of the fact that the structure
constants of a group vary, with the choice of what we may
call the fundamental set of operators, in order to simplify
the structure constants of the group. Thus in § 55 we simpli-
fied the structure of the group of projective transformations
admitted by x2 + y2 + z2 = 1 .
If two groups are such that the structure constants of the
first, corresponding to some one fundamental set of operators,
are the same as the structure constants of the second, corre-
sponding to some one fundamental set of its operators, then
the two groups are said to be of the same structure.
It is, however, a matter of considerable labour when we are
given two groups, with their respective fundamental sets of
operators not given in such a form as to have the same
structure constants, to determine whether or no the groups
have the same structure with respect to some two sets of
fundamental operators.
§ 62. Suppose that we are given a set of structure constants
cikh,... such that all (r—s + l)-rowed determinants, but not
all (r — s)-rowed determinants, vanish in the matrix
cjik> •
cj2k> •
cjrh>'
62]
CONSTANTS OF A GROUP
71
(in any row all positive integral values of,; and k are to be
taken from 1 to r).
We now choose constants a^, ... such that
ahlcjlk + • • • + ahrcjrk ~ °'
(j — l,...,r; k = 1, ...,r; h= l,...,s),
and complete the determination of these constants by taking
amk arbitrarily if m > s ; these arbitrary constants, however,
must be subject to the limitation that the determinant of the
v2 constants
a
IV
a
ru
a
lr
a
rr
=£0.
If a group of the required structure exists, and Xt, ...,Xr
are its operators, then
ahlXl+ ...+ahrXr, (h=l,...,s)
will be s independent operators of the group permutable with
every other operator of the group ; that is, s Abelian operators
forming therefore an Abelian sub-group.
We now take the operators given by (2) of § 61, and thus
we get a new set of structure constants c'ihh, ... with the
following properties :
(«)
ikh
= (L;
ikh
where i, k, h may have any values from (s+1) to r, and
dikh are a set of structure constants of the nth order, n being
written for (r — s) ;
(j3) the constant c'ikh = 0,
if either i or k is less than s+1, h having any value from
1 to r inclusive ;
(y) the constants cikm> ■ ■ ■
where i and k both exceed s, and m does not exceed s, are
such that cikm + ckim= °'
h = r
2 (dikh c'hjm + dkjh cUm + djih c'hkm) = °-
h =s+l
We may therefore say (with the slight change of notation
which consists in writing
dikh = Cr-i, r-k, r-lV and
cikm — ®r-i, r-k, r-m)
72
EXAMPLE
[62
that the problem of finding a group with the required
structure is now reduced to that of rinding a group with
the structure constants d'^, ... defined by the following
properties :
(a) chhh = cikh>
if none of the suffixes i, k, h exceeds n, where the constants
c;j.h are known structure constants of the nth order, such that
not all 7i-rowed determinants vanish in the matrix
cj\lo
(/3) the constant
(3 = l,...,Wx
vfc= 1 n>*
dikh — °>
if either i or k exceeds n, h having any value from 1 to r
(y) d,-hm = dj
vikm
dikm + dkim = 0,
h = n
likm '
where (1)
L ^ (cikh dhjm + ckjh ®>Mm + cjih ^/i&m) = °>
if neither i nor k exceeds n, and m does exceed n.
The constants d^, . . . may be called normal structure
constants, and the problem of finding a group with a given
set of structure constants is now reduced to that of finding
a group with a given set of normal structure constants.
If Ylt ..., Yr are the operators of a group with normal
structure constants, Yn+1, ..., Yr are the Abelian operators of
the group, if any such exist ; and there is no Abelian operator
in the group independent of Yn+l, ..., Yr.
Example.
oe.2, c311 = oCj, c113 = 0, c112 = 0,
ce.-,,
C213 — ~CeZi C312
C223 —
C233 =
^5 ^322 — ^2' ^321 — ae^, C-^o —
C63> C122 —
a63' C332 — 0' C331 —
"» C133 — ^3> C132 — ^2>
C211 — ^61' Clll — ^> C212 — C62' C313
= hev
J1Z\
= o, cm =
c&\i ^222 — "i ^323 — ^^3>
C231 _
aei» C131
O0J, C232 ae2J Cggg 0,
are a set of structure constants, forming the matrix
ce3, 062j ^i>
0, — ae2, —aex,
0, -ce,, 0, —ce2, be.
ce
3'
is
ae
3<
ce2, 0, ce
0, 0, — 6e3, — be2, aev —bev
0, — ae3
ae»
63]
AN IMPORTANT LINEAR GROUP
73
We see that every determinant of the third order vanishes ;
and that, unless a - = b = c, or ex — e2 = e3 = 0, it cannot
happen that every determinant of the second degree vanishes.
If then a group Xv X2, X3 exists with these given constants
as structure constants,
aXx + bX2 + cX3
will be permutable with every operator of the group, that is,
will be an Abelian operator ; and we take then
Y1 = aX1 + bX2 + cX3, F2 = X2, F3 = X3
to be the operators of the group.
We have now a group of which the structure is
(F2 , F3) = 6j Fj + (ae2 - bej Y, + (ae3 - cej F3
(Fl5 F3) = 0, (Fl5 F2) = 0.
If ae2 — bev and aes — ce1 are both zero, we see that Z1 = eF15
Z9 = F2, Z3 = F3 will be three independent operators of the
group with the structure
(Z1,Z.I) = 0, (ZvZ3) = 0, (Z2,ZZ) = ZY.
If ae2 — bel and ae?t — ce1 are not both zero, suppose that
ae2 — be1 is not zero, and take
Z2 = elY1 + (ae2 — bej F2 + (ae3 — ee^ F3, Z3 = (ae2 — be^'1 F3 ,
when we shall have
(Z2,Z3) = Z2, (ZvZ2) = 0, (Zx,Z3) = 0.
§ 63. We have proved in § 58 that the first parameter
group has the same structure constants as the group which
generates it, and that it is a simply transitive group. Now it
may be at once verified that, if
j=h= n ^
*i=2
X
■JM^JTiXi
(i= l,...,n),
then the operators Xj , . . . , Xn, if independent, will form a linear
group with the structure constants c^j., .... The first para-
meter group of this linear group will be simply transitive and
have these constants as its structure constants.
Now the operators Xlt ...,Xn are independent, since by hy-
pothesis not all n -rowed determinants vanish in the matrix
cjrk>
74 EXAMPLE [63
and we thus see that, given the structure constants, the group
can be at once obtained if it does not contain any Abelian
operators.
Example. Find a simply transitive group with the structure
cm=l, cm-0, c211 = — 1, 0^2 = 0, cm = 0,
C112 = 0, (?221 == ^J ^222 = ^'
Writing down the matrix we see that
is a group of the required structure, but it is not simply
transitive.
The finite equations of this group in canonical form are (if
we take e1 X1 + e2 X2 as the general operator of the group)
x'= eeit x+ - (eeit— 1) y, y '= y.
If we change to a new set of parameters given by
ax = ee^, a2 = -1(e^*-l)
62
the finite equations of the group are no longer given in
canonical form, but yet they take the simple form
af=a1x + a2y, y'=y-
The first parameter group is now
x'—axx, y'=a1y + a2,
since the equations which generate it are
Cj = ax bv c2 = bla2 + b2.
The parameter group is therefore a group of the required
type, since it is simply transitive, and it may be verified that
it has the required structure, for its operators are
X f-1/r-) r— •
dx oy dy
§ 64. We now proceed with the theory of the construction
of a group when the assigned structure constants are such
that the group, if it exists, must contain Abelian operators.
64] CONSTRUCTION OF THE GROUP 75
Let Xv ...,Xn be the simply transitive group, which we
have shown how to construct with the structure constants
cikh>"" .
Assuming for the moment that the simultaneous equation
system
h = n
(1) Xi ukm — Xh uim = dikm + 2 Cikh uhm>
(i = 1, ...,n; k = l,...,n; m = n+ 1, ...,r)
can be solved, let ulm, ...,umm be any set of integrals. We can
then at once verify that the r linear operators
J) d
generate a simply transitive group of order r with the structure
constants d'^, ....
Example. Find a group with the structure
I (X25X4) = 0, (Xl5X2) = -X2 + X3, (X1(X3) = 0,
j (X2,X3) = 0, (X3,X4) = 0, (X1,Xi) = 0.
The constants of the proposed group are such that the group
must have two Abelian operators ; and the constants are in
normal form, for X3 and X4 are clearly these Abelian operators.
Using the results of the last example, we take
1 idtfj Jc>a;2 a dx2
and the operators of the required group will be F15 F2, F3, F4,
where 7X = X1 + £3 ^ + £4 ^ F« = *- + *4 +,*4'
F= — , F= —
3 d£3' 4 d#4
We see then, by the condition of the problem, that £3, £4,
tj3, t;4 are functions not involving xz or aJ4, and that
^1 % — ~^2 & = 1 — ^3' ^1 ^4 — ^2 & = °-
76 SOLUTION OF A [64
As we can take any integrals of these equations, we choose
7j4 = £4 = £3 = 0 ; and we must then determine ij3 so that
,3 3 x
[X, r 1- X2 r— ) Tjo = 1 — TJo.
v ^a^ 3av
We therefore take tj.5 = 1 , and we see that
S3 3 3 _*_ _3_
1t)a?1 2 3;r2 3ai2 3#3 3#3' 3#4
will be four independent operators forming a group of the
required structure.
§ 65. We now proceed to show how the equation system (1)
of § 64 may be solved.
Since Xv ...,Xn is a known set of unconnected operators,
- — , . . . , r — can be expressed thus : —
3^ dxn
^--KiXl+-"+XniXm (i=l,...,n),
where \ik , ... is a known set of functions of the variables
Xi , . . . , xn .
From the fact that
3 3 3 3
*zt *xk *xk lx{
and that Xv...,Xn form a group, we see that Xik, ... are func-
tions satisfying the equation system
a =j3 =n
(1) Tx~i-^~Z "j "''
It will now be verified that
a = /3 = n .. -. v
2/ a o o v
dapm (^ Kj Xfik + ^ Kk A|3i + ^- Aai XWJ = 0
I J It
for all values of i, j, k.
We have
3 3 3
^Kj \pk= Aaj — V + A,,— Ki,
3 3 3
^Aa,A,i = A^— A^. + A^— Aa,,
3 3 3
^Ki^j = \ai_Aw + \w_Aai.
65] DIFFERENTIAL EQUATION SYSTEM 77
Since dapm + dPam= 0,
we see that what we have to prove is that
2, Kjdafim (^ Apt— — \f}i) + >, V daPm (j^-Kj — ^7 Ki)
a=P=n ^ ^
+ 2* hpidapm(^ Aai— — — AajJ = 0.
Writing the second and third of these sums in the re-
spectively equivalent forms,
Lhidbpm(—kPj-~\^,
and substituting from (1), we see that the coefficient of
Kj^yi-^bi in the identity is
p =n
— 2 {dpani cybp + ^aM C6y/3 + ^/36m ^ay/3) >
and this is zero by (1) of § 62, so that the identical relation
(2) is now proved.
In order to prove that the simultaneous equation system (1)
of § 64 can be satisfied, multiply the equation there given by
At- A^„, and sum for all values of i, k ; then, if the new set of
equations — there will be one for each pair of values of p, q —
can be satisfied, so can the old.
To see this we notice that for the equation, with a given
pair of values of i, k, the multiplier is A; A^ — Xkp A,- ; and the
determinant of these multipliers cannot vanish, for the deter-
minant of A does not vanish (Forsyth, Differential Equa-
tions, § 212).
If we now take
vim = ^ii uim + • • • + ^ni unm > (* = *»••■» nh
the simultaneous equation system takes the simple form
^ .. i = k = n
V —
vpm — » ®ikm ^ip "kq °^
~bx Qm (>X Pm~ -*-* ikm"tp"kq — "pqm--
where o-f &m, . . . are functions such that
<rikm + (T him = °>
since dikm + dkim= 0;
78 SOLUTION OF A [65
and from (2) we know that
* a d n
*x~t aJkm + toy "**■ + Tx~h *&» - °*
§ 66. To solve these equations consider the following lemma :
if we have -n(n—l) functions o"^,... of the variables
xv ...,xn such that o^ + aki = 0 ,
(i = 1, . . ., n ; ^" = 1 , . . . , n ; A; = 1, . ,. , n),
then n functions uv ...,un can be found such that
To prove that this is true for the case w = 3, let
S2 , d2
oX-^ oX2 "<^i ws
here we can take ux arbitrarily, and obtain u2 and uz by
integration.
*»= ^^(Ul-^}' ** = ^^: (^~Us) ;
Since o"12 + or21 = 0, and o"13 + o"31 = 0,
^2 ,
O"oi =
(U2-Uj), 0-31=___(u3-U1).
21~i)^^2V"2 "1/' ~B1 ^^^
Now ^^+^^1+^^=0, .
a a3
therefore r — <r™ + ^ — =; — ^ — (u., — u9) = 0,
<ix1 li ^x^x^x^ " "
and therefore ^2
°"23 = *x 7>x (u2-u*)+f(x2>x*)-
It is clear that we can write f(x2) x3) in the equivalent form
f(x2 , #,) = r c— (W2 — W~)
where iu2 and w3 are functions of x2, xz only; and if w2 is
taken to be some arbitrary function, then wz can be obtained
by integration ; therefore
32
°"23 =
dar2 ^3
(u2 + w2-u3-w3).
66] DIFFERENTIAL EQUATION SYSTEM 79
Since w2 and iv3 do not involve xl, we see that u15 u2 + w2,
and u3 + w3 are three functions in terms of which a23, <r31> and
o-.^ can be expressed in the required form.
The extension to n variables is now easy. Assuming that
the theorem has been proved for the case of (n—1) variables,
l6t ^ = ^^ (Ui ~ U^ <k = l> "" W>>
where as before ux is arbitrary.
From -^+ — ^ + ±-^ = 0,
we get 4 *** = ^4^K~Wfe)>
and therefore <rM = ^— ^ (uk - uh) + Phh,
where p}:h is a function of x2, ...,xn only.
We have Pkh+Phh = °>
d d d
(i = 2,...,n; h = 2, ...,n; k = 2, ...,n) ;
and therefore, since we now have only (n— 1) variables,
where w2, ..., i0a do not involve xx.
It follows as before that
uv u2 + w2, ..., un + wn
will be a set of functions in terms of which we can express
<rik, ... in the required manner.
If we now write, as we can,
°"-P2TO = *xplxq ^ **" V^'
where the functions V , ... can be obtained by quadrature,
the integrals of the equation system,
■v„m— z — vnm= <Tr
lxp"2™ 7>xq"Pm "PI™
will be vpm = - ^-Vpm.
80
THE FUNDAMENTAL THEOREMS
[67
§ 67. We have thus proved that, given any set of structure
constants, we can in all cases find a simply transitive group
of that structure.
Of the three fundamental theorems in the theory of finite
continuous groups, the first asserts that in a group with
r parameters there are exactly r operators which are inde-
pendent ; and this property, together with the existence of
the identical transformation, is sufficient to ensure that the
equations
will define a group.
The second fundamental theorem asserts that these operators
X
i'
Xr are such that
(xi, Xj) =^cijkX
a;
and that from any set of linear operators satisfying these
identities a group may be generated. The theory of the
canonical form of a group shows us that the group is entirely
given, when we know the linear operators ; and therefore, to
find all possible groups, we have to find all possible sets of
independent operators, such that the alternants of any set are
dependent on the operators of that set.
The third fundamental theorem asserts that this set of
structure constants satisfies the conditions
h = r
cikh + ckih — °>
2* \cikh cjhm + ckjh Cihm + cjih ckhm) ~ ° 5
and that, corresponding to every set of constants satisfying
these conditions, a simply transitive group can be found whose
operators satisfy the conditions
h = r
(X-i> %k) —2*cikhxh-
Later on we shall see how all types of groups with a given
set of constants as structure constants can be found, for so far
the third fundamental theorem has merely shown us that one
simply transitive group of the required structure may be
found.
v{ 83
v function
1 infini-
^ ether
ven
CHAPTER VI
COMPLETE SYSTEMS OF DIFFERENTIAL
EQUATIONS
§68. If q linear operators Xv...,Xq are such that no
identity of the form
cp1 {xv ,.., xn) JL 1 + . . . + <p^ {xv ..., xn) Ay z= 0
connects them, the operators are said to be unconnected.
Any operator which can be expressed in the form
is said to be connected with Xls ...,X ; and all operators so
connected are said to belong to the system Xx, ..., X .
There cannot be more than n unconnected operators, though
there may be an infinity of independent operators ; uncon-
nected operators are of course independent, but independent
operators may be connected (§ 15).
If (f)x (xx, ..., xn) and <£2 (xx, ..., xn) are two functions of the
variables xv ...,xn, such that there is no functional relation
between them of the form
+ (fa, fa) = 0,
they are generally said to be independent ; it will be perhaps
more convenient if we say they are unconnected, and reserve
the word independent for functions not connected by a relation
of the form Kfa + hfa = 0,
where kx and A2 are constants, and not both zero.
Similarly any number of functions fa, ..., </>s will be said to
be unconnected if there is no identical relation between them
of the form ^ (fa, ...,fa) = 0 ■
and they will be said to be independent if there is no relation
between them of the form
^fa+.'. + ^sfa = 0,
where Al5 ...,A4 are constants.
CAMPBELL G
80 THIPLETE SYSTEM OF OPERATORS [68
§ 67. Wave q unconnected operators, such that the alternant
constan4pair is connected with the q operators ; that is, if
ofthr h=q
the operators are said to form a complete system of order q.
If we take any system of unconnected operators Xx, ...,Xq,
and form their alternants (Xit X&), . . .,then, unless each alternant
is connected with Xv ..., X the system made up of XV...,X
and their alternants (X^X^), ... will contain a greater number
of unconnected operators than the original system Xv ...,X
Suppose it contains (q + s) unconnected operators ; we can
add to this system as we added to the original system, and we
shall thus obtain a new system containing still more uncon-
nected operators ; proceeding in this way we must at last
arrive at a complete system, since there can never be more
unconnected operators than there are variables.
If a function of xXi ...,xn is unaltered by the infinitesimal
transformation
xi == ®i ~^'Jti \x\t •••} xn)i v* — Ij ••»} n)>
it is said to admit the infinitesimal transformation, or to be
an invariant of that transformation.
If f (xv ...,xn) is a function admitting this transformation
we must have
i = n ^ .
J \xi j • • • j xn) = J \xl ' • • • ' xn) = J \xl> • • • > xn) + * ^— S« ^T '
it follows that the necessary and sufficient condition that the
function may admit the infinitesimal transformation is that it
should be annihilated by the linear operator
The set of q infinitesimal transformations
xi = xi + t£ki(xV •">xn)> H:-l ' n'
are said to be unconnected if no identities of the form
k = q
2 0ft • £fti («l> — >xn) = °, (*■= 1, ...,'»)
connect them, where (f)v...,(f) are functions of the variables
XD •>•) xn'
n
69]
NORMAL FORM OF SYSTEM
83
§ 69. The problem of finding whether there is any function
/(#!,..., x n) admitting a given set of q unconnected infini-
tesimal transformations, is the same as that of finding whether
there is any function annihilated by each of the q given
operators Y Y
Since, if /is annihilated by Xi and Xj, it is also annihilated
by the alternant (X{, Xj), this problem may be replaced by
that of finding whether there is any function annihilated by
the operators of a complete system.
If the complete system is of order n, i. e. if the number of
unconnected operators is equal to the number of variables,
then the only function which can be so annihilated is a mere
constant.
If, however, the order is less than n, it will now be proved
that there are (n — q) functions which are so annihilated; in
other words, there are (n — q) unconnected invariants of a
complete system of order q.
Let Yv ..., Yq be a new set of operators connected with
Xx, ...,Xq by the identities
Yjc = PklXl+'~+PkqXq> (& = *> ■••)?)>
are any system of functions such that the
where pift, ..
determinant
Pn>
Piq
PqV ' ' ' Pqq
is not identically zero.
The operators F15 ..., Yq also form a complete system of
order q, and any invariant of one system is an invariant
of the other.
In order to simplify the forms of Fj, ..., Yq we now so
choose pik, ... as to have, in the notation of § 38,
i = q
2* Pki %in-q + h — eM'
Since Xl,...,Xq are unconnected, these values of pik, ...
cannot make the above determinant vanish ; we now have
F> =
~dx
n-q + k
G 2
84 REDUCTION OF LINEAR OPERATOR [69
The operators FT, ..., F are now said to be in normal
form, and the problem before us is to find the unconnected
invariants of a complete system given in normal form.
The operators in normal form are all permutable ; for
suppose that
(Yi,Yk) = lx1Y1+...+fxqYq
where fxl} ...,y-q are functions of xx, ..., xn.
From the forms of Fx, ..., Yq we see that the coefficient of
- in the alternant of ( Yj, Yk) is zero ; and, since on the
"xn-q+h
right hand of the above identity this coefficient is nlo we con-
clude that jUj, ..., ij. are each zero.
We now know that Fl5 ..., Yq generate an Abelian group,
all of whose operators are unconnected. (It is not of course
true that the operators XX) ..., X necessarily generate a group;
such a conclusion could only be drawn if XV...,X were
dependent on F15 ..., F„ ; here all we know is that they are
connected with Yx, ..., Yq.)
The problem of finding the integrals of a complete system
of linear partial differential equations is the same as that of
finding the invariants of the corresponding operators ; and
this problem is now reduced to that of determining the
invariants of a known Abelian group, all of whose operators
are unconnected.
It will be noticed that in this reduction of the problem only
the direct processes of algebra have so far been employed.
§ 70. We shall now show how the form of such an Abelian
group may be simplified by the introduction of new variables.
Let x = £i^r +••• + £
~bxx ' *n<)X
n
be any operator, and let fx (xv ..., xn), ...,f]l_1(xv ..., xn) be
any (n— 1) unconnected invariants of this operator, and
fn (xv ..., xn) any other function unconnected with/,, ...,fn_x.
Take as a new set of variables
Vi ==/u •••» Vn =fn '•
then the operator X, when expressed in terms of these new
variables, must be of the form r? r — > where rj is some function
of yv ..., yni which is known, when we know X and its in-
variants.
71] TO SIMPLEST FORM 85
We can now find by quadratures a function $ (yv ..., yn)
such that j. ,
n r — = 1.
This function <£, which we shall now denote by y'a , must con-
tain yn, and must therefore be unconnected with ylt ..., yn_^ •
if then we take as variables yly ...,2/K_15 y'iv the operator X
will be of the form .
o
In order to bring Yq into the form — - , it is only necessary
to be able to find the invariants of a linear operator in
(n—q + 1) variables; for, since the coefficients of
°xn-q+l "xn-\
vanish in Yq, the variables xn_q+l, ...,xn_l can only enter that
operator in the form of parameters.
(It is not to be supposed that in every operator of any
Abelian group the coefficients of r- > •••> r must vanish ;
°xn-q+l °xn-l
but in the particular Abelian group we are dealing with the
operator Yq has this property.)
§ 71. "We shall now prove by induction that every Abelian
group, with q unconnected operators, can be reduced, by a
transformation of the variables, to the form
, . . . .
vXn v%n-i "xn-q+l
Let X15 ...,X„ be the given operators of the group; then
Xl5 ...,X j will form a sub-group of (<?— 1) unconnected
Abelian operators. Assume that these can be reduced to the
forms ^ a
and that *xn-q+i ^»-i
The operators were unconnected and permutable in the first
set of variables, and must therefore retain these properties in
86 REDUCTION OF ABELIAN [71
the new variables ; it then follows that none of the coefficients
£1 in can contain xn_q+l, ..., X^.
By a transformation of the form
2/i =/i (^15 •••> xn-q> Xn)> •••» Vn-q = Jn-q \xl> •••» xn-qi xn)>
yn—fn (xn •••» a'n-g» ^n)' "•' Un-q + l ~ xn-q+\> ""> Vn-l ~ ®n-l*
we can, without altering the forms of Xx, ..., X 15 reduce
X to the form
where £„_2+i, • •, £«-i are functions of yls ..., yB_ffJ yn only.
"We may therefore suppose that Xl , ..., X? have been thrown
into the forms
X - * X - »
where £n_2+1, •••, £»_i do not contain xn_q+1, ...,»„_!; and to
simplify the form of these operators further we take
yl = X1, ...} yn-q = xn-qi Vn ~ xn1>
Vn-q+l = xn-q+\~ I fn-g+l^n' •••' Vn-l ~ xn-i~ Cra-i^n*
We now have
a ^ d d
tyn-q+l *xn-q+l tyn-l *xn-l
and therefore X1, ..., X„ take the respective forms
a 7) a
*yn-q+i' *yn-q+2,'"'}>yn
As we have already proved that any single operator can be
reduced to the form r — > we have now given an inductive
*yn
proof that any q unconnected Abelian operators can, by a
proper choice of variables, be reduced to the forms
CXn oXn_x vxn-q+l
73] GROUP TO SIMPLEST FORM 87
§ 72. When an Abelian group is reduced to this form
xx, ...,xn_q are (n — q) unconnected invariants of the group;
and therefore we have proved that any complete system has
exactly (n — q) unconnected invariants.
It is important to prove that these invariants can be
obtained by direct algebraic processes and integrations of
equations in (n — q + 1) variables at the most.
To prove this we reduce the system to its normal form,
which can be done by processes which are merely algebraic.
If X1 , ..., X are now the operators we reduce X to the form
~)
- — ; this we have proved can be done by quadratures, and
n
the integration of an equation in (n — q+1) variables at the
most.
Xx, ..., X j will now be (q— 1) unconnected Abelian opera-
tors ; let
Xh = ^hi^7 +,,* + £fc»^-' (k=l,...,q—l)
where, since Xk is permutable with — — , £fcl, ..., £kn only
involve x1} ...,xn_v xn
Our object being to obtain the invariants of r- — and
cxn
Xt, ..., Xq_Xi it is only necessary to find those functions
of xx, ..., xn-1 which are annihilated by the (q— 1) linear
operators
414 + - + &»-.^' (*=>••■ •.?-!)•
These (q — 1 ) operators are Abelian operators, and uncon-
nected, so that we have to find the invariants of an Abelian
group in (ft— 1) variables with (q— 1) unconnected operators.
Assuming then the theorem for the case of (n — 1) variables
with (q — 1) operators, we see that it will also be true for the
case of n variables with q operators; and since we have
proved its truth when q = 1, we conclude that the process of
obtaining the common integrals of a complete system of linear
partial differential equations, in n variables, involves the
integration of linear equations in (n—q+l) variables at
the most.
§ 73. Suppose now that we are given the equation
how far are we aided in finding its integrals by our knowledge
88 SOLUTION OF AN EQUATION [73
of (q—1) other operators X2, ..., Xq forming with Xx a com-
plete system ?
We first find the (n — q) unconnected functions which are
common integrals of
X1(f) = 0,...,Xq(f) = 0
by the method just explained ; we then take these functions
to form part of a new set of variables ; and in these new
variables may assume the integrals to be
We now have to find the remaining (q—1) integrals of
where £ls ..., £q are functions of xx, ..., xq, aq+1, ..., an ; the
subsidiary equations of (1) are then
dxx _ dx2 dx,
2
It is known (Forsyth, Differential Equations, §§ 173, 174)
that the solution of these subsidiary equations, and therefore
of the corresponding linear partial differential equation (1),
depends on the solution of an ordinary differential equation
of order (q — 1) in one dependent, and one independent variable.
Thus the solution of £ - f- 77 — = 0, where £ and 77 are func-
dx i>y
tions of x and y, depends on the solution of an ordinary
equation of the first order; £- h 7? f-T— -= 0 depends
^ dx cy dz r
on the solution of an ordinary differential equation of the
second order.
If we define an integration operation of order m as the
operation of obtaining the solution of an ordinary equation of
order m, we may say that : if we are given an equation
X-l(f) — 0, and if we know (q—1) other operators forming with
Xx a complete system of order q ; the solution of the equation
can be made to depend on algebraic processes, on quadratures,
and on integration operations of order (n — q) and (q—1).
Example. Prove that, if X1,...,Xq is a complete system
with the unconnected invariants ux, ...,un_q, then every
operator which annihilates each of these invariants is con-
nected with Xx, ..., Xq.
By a change of the variables we may take the invariants
73] INVARIANTS OF SYSTEM 89
to be icls ..., xn_q ; then the operators are in the variables
xn-q+ i> •••' xn on^7 5 and as ^hey are unconnected
.} ...,
are each connected with Xx, ..., X .
Any operator which annihilates a^, ...,xn_g must be of the
form
and must therefore be connected with Xlt ...3 X .
CHAPTER VII
DIFFERENTIAL EQUATIONS ADMITTING KNOWN
TRANSFORMATION GROUPS
§ 74. In this chapter we shall show how the fact, that
a linear partial differential equation admits one or more
infinitesimal transformations, which may be known by-
observation of the form of the equation or otherwise, enables
us to reduce the order of the operations requisite for the
solution of the given equation.
Let Y be the linear operator
where rjli...,rjn are functions of xv ...,xn, and Y' the operator
obtained from Y by replacing x$ by x\.
(1) If x'^e^Xi, (isl n),
where X = & h... + f»c — »
^xx oxn
we must obtain an expression for Y' in terms of xv ..., a?w,and
this will enable us to determine at once if the equation
Y(f) = 0 admits the transformation (1).
From (1) we deduce (§ 44) xi = e~tx' x\ and therefore
Y'Xi = Y'e~tx ' afj. Since Y'e~tx' x\ is a function of x[, ...,x'n
we therefore have
(2) t,xi = etzYe-txxi.
Expanding etx Ye~tx in powers of t, we see that the
coefficient of V is
Xr7 x'^YX Xr~2YX2 Xr~3YX3
r! (r-1)! + (r-2)! 2! (r-3)!3! + *'"
We shall prove that this expression is equal to (— l)r— -y
where YW = YX-XY, F(2> = Y^X-XY^, ...,
yen _ yv-VX—XYP-v,
75] TRANSFORMATION FORMULA 91
Y(r) having now the meaning which was attached to yr in
§48.
Assume that
F^1) X1-1 Y Xr~*YX Xr~*YX2
I ' (r-1)! == (r-1)! (r-2)! 1! + (r-3)! 2! "'
(-iy(Y(r-vx~XY(r-V)
(r-l)l
XrY X^YX Xr~2YX2
~(r-l)! (r ^(r-l)!l!+'r ')(r-2)!2! '"
Xr-iTX Xr~2YX2
+ 2
then
(r-1)! 1! (r-2)! 2!
XT P-TI Xr~2YX~
— ** ; , ,. - + r
da^
"(r— 1)! (r— 1)! 1! (r-2)! 2!
and therefore
YW_XrY _XT-1TX Xr~2YX2
* ' r! ~~ r! (r-1)! 1! + (r-2)! 2! ""
so that the required theorem is proved by induction ; and
etxYe~tx = Y-tYW + ^.YW-^,YW + ....
— ! o I
It follows that etxYe~iX is a linear operator, and as such
it may be written in the form
and by (2) this may be written
i = n
■^ v l/ <>a^
so that F'= F- 1 FW + |j Y& - ^ F(3) + . . . .
§ 75. We may apply this formula to obtain the conditions
that a given sub-group may be self-conjugate.
If Xx, ...,Xn are the infinitesimal operators of a group, of
which X +1, ...,Xr form a sub-group, we defined a self-con-
jugate sub-group as one such that
is always an operation of the sub-group, whatever be the
92 CONDITION THAT A SUB-GROUP [75
values of ex, ...,er, the parameters of the group, or Xq+l, ...,Af
the parameters of the sub-group.
If we denote by X the operator e1X1+ ... +erXr, this con-
dition may be expressed by saying that the group generated
by X'q+l,...,X'r, where
/ Y
'"t — C *°IJ
is identical with the group Xq+l, ...,Xr; that is, that each
operator X'q+l, ...,X'r is dependent on the operators of the
Set -A q+i, ..., Ar.
Now the formula we have just proved gives
X>k = Xk-XV + ±Xf-±Xf + ..., (* = g + l,...,r),
so that
must be dependent on Xq+1, ...,Xr.
By the second fundamental theorem (§ 47) we have
i = k = r
and therefore, if we take el,...,er so small that their squares
may be neglected, we see that a necessary condition for
Xq+1, ...,X'r being dependent on Xq+1, ...,Xr is
Since this must be true whatever the values of the small
quantities e1,...,er we must have
The sub-group X„+1,...,Xr cannot then be self-conjugate
unless these conditions are satisfied.
These necessary relations between the structure-constants
are also sufficient ; for if they are satisfied Xq\j will be
dependent on Xq+1, ..., Xr ; and therefore, since this is true
for all values of j from 1 to r-q, Xf+j, Xf+ji ... will all be
dependent on Xq+1, ..., Xr, and therefore X'+j will be so
dependent.
.
76] MAY BE SELF-CONJUGATE 93
If we take q = r— 1, we get in particular as the conditions
that Xr may be a self- conjugate operator
/i — 1, .... r n
If Xr is to be an Abelian operator the further conditions
crir = Oj (^ = 1) •••} T)
are necessary.
§ 76. We now seek the conditions that the complete system
of equations *■,</)_ 0, .... Ff </)= 0
may admit the group of order one
Clearly the conditions are that Y[,...,Yq should each be
connected with Tx , . . . , Yq ; that is, we must have
Yk = PklYl+---+PkqYq> (k=l,...,q),
where pki, ... are functions of #15 ..., xn.
Since n=F,-^Y + ^Ff-...,
we see, by taking t very small, that necessary conditions are
where <rfci, ... are some functions of xx, ...,xn.
These necessary conditions are also sufficient ; for
Yf = (^1^1+...+^ Yvx) = Tto^p+.-.+o^rw
+(^i)*r1+...+(X(r&2)rg,
and therefore, since Fp, ..., F^ are each connected with
Fl5 ..., F , we see that Y^'f is also connected with F15 ..., F„.
Similarly we see that F^, F^, ... are each so connected;
and therefore Y[, ..., F' are connected with Fl3 ..., F • and
we conclude that the necessary and sufficient conditions that
a comjilete system of linear partial differential equations of
the first order should admit the group
x^ = e x^ , yi — i , . . . , n)
are that the alternants (F1? X), ..., (F„, X) should each be
connected with YXi.,.,Yq.
94 EQUATIONS ADMITTING KNOWN [77
§ 77. If /(aJj ic„) = constant is any integral of the com-
plete system, that is, if f(x1} ..., xn) is any invariant of the
complete system of operators Fl5 ..., Yq, then /(^i,...,*^) is
an invariant of Y[, ...,Y' Now by hypothesis the complete
system admits j _ etx x,
and therefore by what we have just proved
Y'k = PklYl+ — +PkqYq> (k= 1» •••»?).
The determinant of the functions pik , ... cannot be zero ;
for if it were zero Y[, ..., Y would be connected, and there-
Y1,...,Yq (being operators of the same form, but in the
variables x1} ...,xn instead of x[,...,afn) would be connected,
and this is contrary to hypothesis : since then the determinant
is not zero, every invariant of Y[, ...,Y' is an invariant of
Fl5 ..., Yq; and we conclude that \if(x1, ..., xn) is an invariant
of Yl, ..., ¥„ so also \% f (x'x, ... ,x'n).
In other words, any invariant of the complete system of
operators is transformed by
x^ = e Xj, yi = 1, ...,n)
into some other invariant function, if the complete system
admits this transformation.
We may prove conversely that if
of. = etxx{, (i = l,...,n)
transforms every invariant of the complete system into some
other invariant, then the complete system admits this trans-
formation.
For suppose that f(xlf ..., xn) is an invariant: then by the
hypothesis so is f(x[, ..., x'n), that is
e J \**i > • • • ' *"«/
is an invariant. If we now take t very small, we may con-
clude that Xf '(x1, ...,xn) is an invariant, and therefore must
be annihilated by Fls ..., Yq.
Since f(xl,...,xn) is an invariant, it is annihilated by
F15 ..., Yq, and therefore also by the operators of the second
degree AT,,... JF ■ and therefore finally f(xu ..., xn) is
annihilated by each of the alternants (Fl5 X), ..., (Yq, X).
It follows then from the example on page 89 that each
of these alternants is connected with Fl5 ..., Yq, and therefore
that the complete system admits
x\ = etxx{, (i=l,...,ri).
78] INFINITESIMAL TRANSFORMATIONS 95
We thus see that the conditions that a complete system may
admit the above group may be expressed by either of two
equivalent conditions ; firstly, by the condition that the alter-
nants of each of the operators of the complete system with X
should be connected with the operators of this system ;
or, secondly, by the condition that every invariant of the
system should be transformed into another invariant by the
operator X.
§ 78. The condition that a given function f(xx, ...,xn) may
admit
(1) x$ = Xi + iQi (xx, ..., xn), (i=l,...,n)
is that it should be annihilated by the operator X,
where X = £x - h . . . + £
~bxx ' ' n <iXn
It must therefore, if it admits (l), also admit
(2) x'i = xi + tpgi(x1,...,xj, (i=l,...,n)
whatever function of the variables x1,...,xn the multiplier
p may be.
If on the other hand a given differential equation Y(f) — 0
admits (1), it will not in general admit (2).
If Yx{f) = 0,..., Yq(f) = 0 is a given complete system
of differential equations the system will obviously admit the
infinitesimal transformation.
(3) afi = xi + t(p1Y1 + ... + PqYq)xi
whatever the functions px, ..., pq may be ; for the alternants
of Y1,...,Ya with p1Y, + ...+paYa are connected with
Y Y
A transformation of the form (3) is said to be trivial.
If the equation system admits
■*"* — e ^ii
we say that it admits the operator X ; and we now see that if
it admits X it will also admit
X-\-PlYx + ... + pgYq\
but with respect to the given equation system we should not
reckon x\ = e^^
and afi = etr+f^Yi + "-+P9Tixi
as distinct transformations.
96 EQUATIONS ADMITTING KNOWN [78
We can, however, make use of the fact that px,...,pq are
undetermined to obtain the simplest forms of the operators
admitted by the given equation system.
Suppose that the complete system admits the non-trivial
transformation . _ „,*£/„ „ \
•^i — ^% t t- c,% \^i > '"> ^n/>
under what conditions will it admit
x'i = xi + tp£i(x1,...,xn)'l
The conditions are that the alternants (F,,pX), ..., (Yq,pX)
should each be connected with F15..., Yq; and therefore,
since p ( Y1 , X), . . . , p ( Yq , X) are each so connected,
(YlP)X,...,(YqP)X
must each be connected with Y1 , . . . , Yq .
Now by hypothesis X is not connected with F2 , . . . , F ; and
therefore we must have
YlP = 0,...,YqP = 0;
that is p is either a constant, or an invariant of the complete
system.
§ 79. If the complete system is reduced to normal form,
that is if
CU/n-q + k 0dji
the further discussion of the problem with which we are now
concerned is made more simple. This problem is the in-
vestigation of the reduction of the order of the integration
operations, necessary for the solution of the given equation
system, due to the fact that the system admits known non-
trivial transformations.
Since the reduction of the system to normal form only
involves algebraic processes, we may suppose the system to be
given in normal form.
If X is a non-trivial operator admitted by the system, then
X + plY1 + ...-vPqYq
is also admitted, and is non- trivial ; and, by properly choosing
the functions px, ...,p we can replace X by a linear operator
of the form d d
which is necessarily non-trivial.
80] INFINITESIMAL TRANSFORMATIONS 97
We shall call such an operator a reduced operator ; and
when we are given any non-trivial operator admitted by the
system, we replace it — and this can be done by mere algebra —
by the corresponding reduced operator.
If then we are given a complete system, in normal form,
admitting m known unconnected reduced operators X15 ..., Xm
we must have
(%{* Yk) as (r1Y1+ ... + <rqYq.
Now in (X{, Yk) the coefficients of r > •••> r — are all
zero, and therefore we must have ax = 0, ..., crq = 0 ; each
of the operators Xx, ..., Xm is therefore permutable with
each of the operators Y1, ..., Yq. Also there cannot be more
than (n— q) reduced unconnected operators Xx, ...,Xm, for
these operators are in the (n — q) variables xx, ...,xn_q only,
xn-q+i> •••>a?n entering them merely as parameters.
We also see as in § 78 that
PXXX + ...+pmXm
can only be admitted if Pi,...,pm are invariants of the
operators Yx, ...,Yq.
From the Jacobian identity
(F&, (X„ Xj)) + (Xj, (Yk, X{)) + (Xi} (Xj, 7k)) = 0,
we see that, since (Ffe,Xt-) and (Yk, XA vanish identically, so
also must (Yk,(Xi,Xj)); that is, the equation system admits
the alternant of any two reduced operators ; and this alternant
is itself a reduced operator since it is of the form
t J_ t
**l "^^n-a
It therefore follows that, if an equation system admits any
non-trivial operators at all, it must admit a complete system
of operators; we shall suppose then that Xlt ...,Xm is a
complete system of operators in the variables xx, ...,xn_q, the
other variable xn_q+l, ...,xn entering these operators only as
parameters; and we know that m>?i — q.
§ 80. We now have
(%i , Xj) — Piji ,Xx + ...+ Pijm Xm ,
and, since the system admits (Xi,XAi the functions p^k> ...
are either constants, or integrals of the given equation system.
CAMPBELL 2
98 EQUATIONS ADMITTING KNOWN [80
The first thing which we must now do is to reduce the case
where the functions are integrals to the case where they are
mere constants.
Suppose that of the functions pi -^ , ... exactly s are uncon-
nected ; we now know s invariants of the complete system,
and we therefore transform to a new set of variables, so chosen
that xn_q,xn_q+1, ...,xn_q_s+1 are these known invariants of
the complete system.
This transformation of the variables has only involved
algebraic processes ; and we now again bring the system to
normal form, when we have
i = n — q — s
CUyn-q + k OJbi
We suppose Xx , . . . , Xm , the operators which the equation
system admits, again reduced, so that
i = n—q
** = 2.&^-» (&=l,...,m).
if
From the fact that (Yi; X^) = 0, and that none of the terms
occur in F15 ..., YQ, we see that
o^n-q-s+l VJjn-q
*>&».- o, (;:":?7+1' ■•••*"')•
It therefore follows that £j.^, ... are integrals of the system:
they may either be new integrals or they may be connected
with the known set xn_q, ...,xn_q_s+1.
If they are new integrals we simplify Y1, ..., Yq still further
by again introducing the new integrals as variables ; and
continue to do this till we can obtain no further integrals
by this method.
We may therefore now assume that
=kh>
(Ji — n — q — s+1, ...,7i — q)
are merely functions of xn_q, ..., xn_q_g+1, that is, of the
integrals already known.
§ 81. It must be noticed that we cannot advance further in
obtaining integrals of the complete system, through our
knowledge that the system admits XXi ...,Xm, unless in so
82] INFINITESIMAL TRANSFORMATIONS 99
far as we know how to deduce from X1,...,Xm operators
of the form »=»_«_«
To prove this, suppose that the system admits X which
is of the form ,=*_,_,
+ 2 &
We now have the complete system of equations
X(f) = 0,Y1(f) = 0,...,Yq(f) = 0,
and it is in normal form ; but, since we have increased the
number of the variables as well as of the equations, the order
of the integration operations, necessary to find a common
integral, is now no lower than it was to find a common
integral of 7l(/) = o,...,r,(/) = 0.
We take
zh = Phi ^i+...+ P]cmXm> (fc= l,...,m),
where pki,... are functions of xn_q, ...,xn_q_s+l only, and
are therefore invariants of Yli ..., Yq. Zl3...t Zm will now be
reduced operators admitted by the given equation system.
We must so choose p^, ... as to obtain as many as possible
of the operators in the form
tl N _ + ... + C?i-o-j
and these alone can be effective for our purpose.
§ 82. The problem before us is now simplified and may be
thus restated : we are given q operators F15 ,.., Yq where
i = n-q-s
Tk = sz — i +2 ^-^-.' (* = *« ■••'?) 5
°xn-q + h cxi
and, in order to obtain new integrals of the system, we are to
make the most use of our knowledge that the system admits
Xlt ...,Xm where
i=n—q—s ..
h a
100 EQUATIONS ADMITTING KNOWN [82
As before we have
(Xj , Xj) = pjjX Xl + . . . + pijm Xm ,
and the functions p#t, ... being invariants, we should have
new integrals unless they are merely functions of the known
integrals xn_q, ..., xn_q_s+1.
Since we have assumed that we cannot obtain any more
integrals by this method we must take these quantities
Pijk,... to be merely functions of xn_q, ..., xn_q_s+l ; and,
since these variables only enter Ylf ..., Yq, XXi ..., Xm as
parameters, we may now assume p{jk, ... to be mere
constants.
The operators Xlt ..., Xm then satisfy the identities
(r*Jj) -2 «*»*». (J"!'--").
"* J VJ = 1, ..., my
that is, they generate a group.
We thus see how Lie's theory of finite continuous groups
had its origin in the question which he proposed, viz. what
advance can be made towards the solution of linear partial
differential equations of the first order, by the knowledge of
the infinitesimal transformations which the equation admits ?
§ 83. We know that (m + q) is not greater than n ; suppose
that it is less than n. We then find the common integrals of
the complete system
X,{f) = 0 Im(/) = 0, Yx(f) = 0, ..., Yq(f) = 0,
of which all the operators are unconnected, and of which the
structure of the operators — for these operators generate a
group of order (m + q) — is given by
(X{, Xk) = ci}il X x + . . . + c i]imXm ,
and by the fact that the operators Yx, ..., Yq are Abelian
operators within the group of order m + q>
There are (n — m—q) common integrals of this system which
can be found by an integration operation of order (n — m — q).
Having determined these integrals we so change the variables
that the corresponding invariant functions become
#ti> •«•> xm + q + l '
and the problem of finding the remaining integrals of
F1(/) = o,...JF2(/) = o
85] INFINITESIMAL TRANSFORMATIONS 101
is now reduced to that of finding the invariants of a complete
system of order q, in (m + q) variables xx, ..., xm+q, the system
admitting ra known reduced unconnected operators, also in
the same variables xx, ...,xm+q.
As (m + q) is either less than n or equal to it, we can now
restate the problem in the form to which we have reduced it.
Given a complete system of equations
Yx(f) = 0,...,Yq(f) = 0
in (r + q) variables xx, ...,xr+q, whose invariants are required,
we are to take advantage of the fact that the system admits
r known operators Xx, ..., Xr in these variables.
The r operators are unconnected, and reduced, and generate
a group which is finite and continuous ; and the variables
xn, ...,xn_r_q+1 occur in Xx, ...,Xr, Yx, ..., Yq, merely as
parameters; Ylt ..., Yq are operators permutable with each
other and with Xx, ..., Xr.
§ 84. In order to find the invariants of Yls ..., Yq we should
have required integration operations of order r, had it not
been that we know that the equation system admits the
operators Xx, ..., Xr . We therefore find the maximum sub-
group of Xx, ..., Xr ; that is, the sub-group with the greatest
number of independent operators, which being a sub-group
must not include all the operators of the given group
Xx, ...,Xr; and we find the integrals of the system
Yx{f) = 0, ..., Yq(f) = 0, Xx(f) = 0, ..., Xm(f) = 0,
where Xx, ..., Xm is this maximum sub-group.
To obtain these integrals, integration operations of order
(r — m) are required, and (r— in) integrals are thus obtained;
the reason why we choose m as large as possible is to reduce
the order of the necessary operations ; and the reason why we
choose a sub-group is to ensure that (r—m) shall not vanish.
We shall now show how, by merely algebraic processes, we
may obtain other integrals from these (r — m) integrals.
§ 85. The principle which enables us to find these additional
integrals is that explained in § 77. Since the given system
admits X1} ..., Xr, we know that if $ (xx> ..., xn) is any
invariant of Yv ..., Yq, then Xx<f>, ...,Xr<f> will also be in-
variants. All of the invariants we have already found can
be annihilated by Xx, ..., Xm ; but they cannot all be annihi-
lated by Xm+X, nor by any of the operators Xm+2, ...,Xr; we
102 EQUATIONS ADMITTING KNOWN [85
may therefore by this method be enabled to obtain new-
integrals.
By a change of the variables, that is, by an algebraic
process, we may take the invariants already known to be
xr+q> ••'' q + m + l'
Let X1; ..., Xi be that maximum sub-group of Xlt ..., Xm
which is self-conjugate within Xls ..., Xr ; if Xls ..., Xm is
itself self-conjugate within Xls ..., Xr, we may take Xl5 ..., Xj
to be the sub-group Xlt ..., Xm itself.
The proposition which we are now going to establish is
this — by operating with Xu ...., Xr on the known invariants
xr+Q, ..., xq+m+1 we obtain the common integrals of
¥,(/) = 0, ..., Yq(f) = 0, Xx{f) = 0, ...,*,(/) = 0;
that is, we obtain exactly (m — l) additional integrals.
Since all of the variables xr+q, ..., xq+m+1 are invariants
of Fl5 ..., Yq, Xj, ..., Xm they must also be invariants of
Fl5 ..., Yq, Xls ...,Xi\ by a change of variables we may take
xq+mi ••■> xq+l+i ^° De ^ne remaining invariants of
V V Y Y .
we are now about to prove that by performing known opera-
tions on xr+q, ..., xm+q+1 we must obtain these additional
invariants.
Since Xls ..., Xj is a self-conjugate sub-group of X15 ... Xr,
the equations
Xx(/) = o, ...,It(/) = o, Yx(f) = o, ..., Yq(f) = o
admit the operators X1, ...,Xr; and therefore the functions
obtained by operating with X1,...,Xr on xr+q, ...,xm+q+1
must all be invariants of X1? ..., Xj, Fl5 ..., Yq.
Now Xls...,Xj, Y1,...,Yq are unconnected, and have as
invariants the (r — l) variables xq+i+1, ...,xr+ ; every other
invariant must therefore be a function of these variables only ;
and therefore we know that the invariants obtained by
operating with XlJt..,Xr are functions of xq+i+1, ...,xr+q
only.
If (r — I) of these invariants are unconnected, then
xq + l + l> "•■>xr + q
can be expressed in terms of these invariants ; but if fewer
than (r—l) of the invariants are unconnected, they cannot be
so expressed ; and we therefore know that there must be some
operator of the form
Sq + l+l^r +'"+Sq + m^
85] INFINITESIMAL TRANSFORMATIONS 103
which annihilates each of the functions Xixq+m+j, where j
may have any value from 1 to (r—ni), and i any value from
1 to r, and where £q+i+i, ...,£g+m are not all zero.
Since xq+m+l, ...,xr+q are invariants of
7 V Y Y
l» •••»■* o> -^-ij • ••} -^-m'
and these operators are unconnected, we see that
■ j • • • »
must be connected with Y1,...,Yq, X15...,Xm; we can
therefore replace
°^2+? + l ox<i+m
by an operator of the form
where p1} ..., pq, o-ls ..., crm, are functions of the variables.
Now each of the operators Yx, ..., Yq, Xx, ..., Xm annihilates
each of the variables xq+m+x, ...,xr+q, and (1) annihilates any
function Xixq+m+j ; we conclude then that
Pl(Y1, Xi)+-..+Pq(Yq, xj + cr^x,, x,.)+...+^(^m, a;.)
annihilates each of the variables xq+m+x, ...,xr+q.
From the known relations between the alternants of the
operators Yx,...,Yq, Xx, ..., Xm we see that
j=m, k=r
!L<rjCjikXlc> (i=l,...,r)
annihilates each of these variables ; and must therefore be
connected with the operators of which xq+m+x, ...,xr+q are
the invariants; that is, with Yx, ..., Yq, Xx, ...,Xm.
It follows that, these operators being all unconnected, we
must have
i = m .
Now because Xx, ...,Xl is a self-conjugate sub-group
= 0, (* "]""'''' k = l+l,...,r);
KJ = 1, ...,t; J
and therefore
}=m ' __ 1
2 »i «/» = ?» (l = m + i,...,r)
104 EQUATIONS ADMITTING KNOWN [85
Xls ...,Xm is a self-conjugate sub-group, and I — m. If m>l,
these constants cannot all vanish (for then the greatest
sub-group would be of order > I) ; and we can take one of
the functions 07 +1, •••>o"m to be dependent on the others; it
follows that without altering the structure of Xly ...,Xj, or
without transforming the sub-group Xx , ..., Xm into any other
sub-group, we may choose instead of Xj+1, ..., Xm certain
(m — l) independent operators which will be dependent on
Xi+1, ..., Xm, and for this new set we may take <rm to be zero.
If we now consider the corresponding new structure con-
stants, we shall as before obtain the identities of the form
j = m-l
,„+?Ci!'* (* = ™ + l r><
and can similarly choose <rm_j to be zero, and, proceeding thus,
finally cause all the functions o-j+1, ...,crm to disappear.
It would then follow that
r ••• + tflf+i
OJjq + l + l 0tlq+m
could be replaced by an operator of the form
(2) p1Y1+...+PqYq + <rlX1+...+*lXl;
but this is impossible since (2) annihilates xq+i+l, ..., xq+m:
we must therefore draw the conclusion that xq+i+1, ...,%q+m
can be expressed in terms of the invariants obtained by
operating on the known invariants %q+m+1, ...,%r+q with
§ 86. It therefore follows from what we have proved that
we can by an integration operation of order (r — m) obtain
(r — l) invariants of Y1, ..., Yq; and we may take these to be
xr+q> •••> xq+i+i> ty a transformation to new variables.
The variables xr+q , ..., xq+i+l now appear only as parameters
in Fl5 ..., Yq ; we can therefore, by processes which are merely
algebraic, select from the r operators Xls ...,Xr which the
equation system admits I operators, in which also
xr + q> •••> xq+l+l
will only appear as parameters. These will form a group of
87] INFINITESIMAL TRANSFORMATIONS 105
order I in (I + q) variables, and will be unconnected with one
another, or with Tlt ..., Yq. The equation system
F1(/)-o,...lra(/)-o
will admit these operators, and the problem which is now
before us is exactly the same as it was before, but we have
only (l + q) variables to deal with, whereas before we had
(r+q)>
§ 87. There is one case of special interest in this general
theory, viz. when the greatest sub-group of X1} ...,Xr is self-
conjugate.
Since Xx, ..., Xm is self-conjugate, the alternant of any of
these operators with Xm+l is dependent on X1? ...,Xm ; and
therefore Xl5 ...,Xm+l is itself a sub-group; but Xx, ...,Xm
is by hypothesis the maximum sub-group, and therefore
Xj, ...,Xm+l must be the group Xx, ..., Xr itself.
When the greatest sub-group of Xlt ...,Xr is self-conjugate
its order must therefore be (r— 1).
There is only one invariant of F15 ..., P_, Xlt ..., Xr_x ; sup-
pose it to bef(xls ..., xr+ q), then, since Xr (/) must also be an
invariant,
where F is some functional symbol.
This function F(^f(x1, ..., xr+q)) cannot be zero ; for
FY Y Y
being unconnected have no common invariant ; there must
therefore be some function of f{xli ..., xr+q), such that, when
operated on by Xri the result will be unity.
Let u be this required function, then
71(u) = 0, ...,Yq(u) = 0, X1(u) = 0,...,Xr_1(u) = 0,
Xr(u) = 1.
Since these are (r + q) unconnected equations in (r + q) variables
every derivative of u is known: that is, - — , ••-, are each
known, and u can therefore be obtained by mere quadrature.
By transforming to a new set of variables we may take this
function to be xq+r; since xq+r will then occur merely as
a parameter in Yx, ...,Yq, Xx, ...iXr_1 we shall then be given
an equation system
Y1(f) = 0,...,Yq(f) = 0,
106 LIE'S METHOD OF SOLVING [87
in (r + q— 1) variables which will admit the group Xlt ...jX,.,^
and Xlt .... Arr_15 Y1, ..., Yq will all be unconnected operators.
If the greatest sub-group of A^, ..., Xr_1 is self-conjugate,
we may take this sub-group to be X1} ...,Xr_2, and thus by
quadratures obtain another integral of
F1(/) = 0,...,Fg(/) = 0;
and hence proceeding find all the integrals by quadratures,
provided that each successive maximum sub-group is self-
conjugate within the previous one.
§ 88. Suppose we are given the linear differential equation
how far does the method explained help us in obtaining some
or all of its integrals ?
We know that by a suitable choice of variables the equation
■\J!
may be reduced to the form ~- = 0 ; and therefore it will
admit any operator whose form in the new variables is
where tj2, ...}i?„ are functions of x2, ..., xn only. Every equa-
tion must therefore admit (n— 1) reduced unconnected
operators ; but, since the reduction of a given equation to
the form xr~ = 0 would require integration operations of
cx1
order (n— 1), we do not know any general method of obtain-
ing the infinitesimal operators admitted by the given equation.
Lie's method does not therefore apply to any arbitrarily
chosen differential equation, but merely to those equations
which admit known operators. These operators may be known
from the form of the differential equation, or from its geo-
metrical genesis.
When we do know, by any method, the integrals of a given
equation, it would be a simple matter to construct infinitesimal
transformations which the equation will admit ; and then,
knowing these infinitesimal transformations, we could solve
the equation by Lie's method. Such examples would how-
ever merely serve as exercises in applying the method, and
could not show its real interest. What is remarkable is that
those particular types of differential equations whose solutions
90] DIFFERENTIAL EQUATIONS 107
have long been known, and were discovered by various arti-
fices, are equations which do admit obvious infinitesimal
transformations, i.e. transformations which would be antici-
pated without any knowledge of the solution of the equation
and merely from its form, or from the geometrical meaning
of the equation.
§ 89. Before illustrating the method by a few simple
examples it will be necessary to consider how it applies to
ordinary equations in two variables.
Consider the equation
(i) yn+i=f(x>y>yi>—>yJ>
oV v
where yr is written for -j— •
Since dx = dy = d]Ll=^= dyn
we see that the solution of (1) will be obtained only when
we have obtained all the invariants of
3 3 3 . 3
c — hy-,^ — i- y->rr— + ... +fz j
3a; yily yilyi Jlyn
x, y, yx, ..., yn being regarded as unconnected variables.
If the equation (1) admits
x'=x + t{(x,y), y'=y + tri(x,y),
then we have shown how to extend this point transformation
to any required order ; and therefore corresponding to any
known infinitesimal transformation admitted by
y»+i=/(«»y»yi»— »y«)
we shall have a known infinitesimal transformation admitted
by <>u 3u „3w
lx &1ly J lyn
and we can therefore reduce the order of the integration
operations necessary for the solution of (1).
§ 90. We shall now give one or two simple examples of
the application of Lie's method.
Example. Consider the linear equation
yi+yffc) = <t>(x)>
where yx is written for -~ •
108 EXAMPLES IN ILLUSTRATION [90
Let any integral of this equation be y = £, where £ is a
function of x, and let 2/ = £° be any integral of yi + yf(x) = 0,
then y = £ + c£°, where c is an arbitrary constant, is also an
integral; we express this in Lie's notation by saying that the
given equation admits the infinitesimal transformation
y'=y + t£°, x'=x.
The partial differential equation
ou / , . . .. .v ou
therefore admits the operator £° — ; and, if u is any invariant
is
of ^- +(0(«) — ?//'(#)) c-» then £0— will also be an invariant,
and will therefore be a function of u.
We can then find some invariant v, such that
j- + (#(*)-tf<*))^-0, f.^-1,
and such therefore that — and — are known in terms of
ox oy
x and y. We can therefore find v by mere quadratures, and
thus deduce the complete primitive from our knowledge of
two particular integrals, viz. one of the equation
£+yf(x)s=<j)(x))
and one of the equation
Example. The equation
obviously admits the transformation
x'= ax, y'= ay,
where a is a variable parameter, and therefore
ox J ^ xJ oy
91] OF THE METHOD 109
admits x - — \- y — •
so that the homogeneous equation of the first order can be
solved by quadrature.
Example. Curves whose equations are given in the form
of a relation between r and p, where r is the distance of
a point on the curve from the origin, and p the perpendicular
from the origin to the tangent at the point, can always be
solved ; that is, we can obtain the Cartesian equation of these
curves. These equations are of the form
y-m= ^i+2/i2/(*2+2/2).
and, from their geometrical meaning, must be unaltered by
rotation of the axes of coordinates ; that is, they admit the
operator y x — - and can therefore be solved by quad-
ratures. ** *
§ 91. Euler has shown how to integrate the equation
b + ex + gy + hxy + ky2 m
^1~ a + cx + dy + hx2 + hxy '
we shall show how this would be solved by Lie's method.
Writing down the equation
(a + cx + dy + hx2 + kxy) — +(b + ex + gy + hxy + Jcy2) — = 0,
oqo y
we are to find some infinitesimal transformation which it
will admit.
It is obvious that any projective transformation must trans-
form this equation into another of the same form, though not
necessarily with the same constant coefficients a, b, c, d, e,f,
g, h, lc\ we therefore seek that particular projective trans-
formation (if such exists) which the equation may admit.
It is now necessary to state a general theorem (the proof
will be given later) which will help us in finding the forms
of the infinitesimal transformations which a given complete
equation system may admit.
Suppose that Yx(f) = 0, ..., Yq(f) = 0 is a complete equa-
tion system of order q and that
XT * *
yj*-Vkl*x~+-+TlknZx~n'
110 POINTS OF SPECIAL POSITION
then not all (/-rowed determinants of the matrix
[91
»?it>
Vqi:
Vm
Iqn
can vanish identically.
A point xl,...,xn such that, when we substitute its co-
ordinates in the matrix, not all (/-rowed determinants of the
matrix vanish, is said to be a point of general position ;
a point such that all (h + l)-rowed determinants, but not all
/(-rowed determinants vanish, is said to be a point of special
position of order h ; h may have any value from 1 to q, but
if h is equal to q the special point becomes a general point.
The theorem, assumed for the present, is that by any trans-
formation, which the given equation system can admit, a point
of general position must be transformed into a point of general
position ; and a point of special position into a point of
special position of the same order.
In the example we are considering the points of special
position are those points which satisfy the two equations
a + cx + dy + hx2 + kxy = 0, b + ex + gy + hxy + ky2 = 0.
We see that in general there are three points not at infinity,
and one point at infinity, common to these two conies ; by
a linear transformation of coordinates we may take these
points to be the points whose coordinates are respectively
(0, 0), (0, 1), (1, 0),
and in this system of coordinates the equation whose solution
is required is
(1) {ch{x-x2)-a2xy)~ +(a2(y-y2)-a1xy)^ = 0.
Since we are now seeking a projective transformation which
the equation will admit, it must be one which will not alter
the points (00)] (01)j (I0)i
and it will therefore be of the form
{aAx-x^-a^y)—-^ (a2(y-y2)-aixy)^,
where ax, a2 are undetermined constants.
92] EXAMPLE 111
We now easily see that the equation (1) admits
{x-x*)~-xy^ and (y-y*)^--xy±.
These two operators are not reduced unconnected operators,
but the knowledge of either is sufficient to reduce the solution
of (1) to quadratures.
As our object is to illustrate the uniformity of Lie's method
as contrasted with the earlier and more special methods, and
not actually to obtain the integrals of differential equations,
we shall not carry out the operations necessary to obtain the
explicit solution of the equation. It may often be found
that the special methods with which we are familiar will
obtain the solution of known equations more rapidly than
we can obtain them by the more general method of Lie.
§ 92. As an example of Lie's method of depressing equations,
take the known result that a differential equation can be
depressed when one of the variables is absent. Since, if x
does not appear in it, the equation must admit — > and if y
does not appear it must admit — > we see that the integration
operations necessary for the solution are lowered by unity.
So if neither x nor y occur explicitly the order may be
depressed by two, for the equation will now admit — and — •
Again, any homogeneous equation can be depressed since it
o o o
admits x z — \-y^ — bz~ + —
ox oy oz
Thus, if we take
, v . , , ou , 7 . ou
(1) (a1x + b1y + c1z) — + (a2x + b2y + c.,z)^
+ (a3x + b3y + c3z)~ = 0,
since it admits x z — \-y^ — Yz — > we must find the common
dx oy dz
integral of (1) and
ou ou ou
ox oy oz
112 EXAMPLE [92
eliminating — this common integral must satisfy the equation
z(a1x + b1y + clz)—+z(a.,x + b2y+c2z) —
= (v+ hy + c*z)yx Yx+yMj)'
In this equation z occurs only as a parameter, and therefore
taking x = xz, yf — yz, the equations become
(a.x' + b.y +Cj)^ + (a,a +622/ +^2)^>
We have proved that the integral of this can be obtained
by a quadrature ; and therefore u must be of the form
F(x,y,z) + (f)(z),
where J7 is a known function and <£ (z) an unknown function.
Since u is annihilated by x— +Vz — K— the unknown
* ox oy dz
function <f) (z) can also be obtained by quadrature.
Having thus obtained the common integral of the equations,
we introduce it as a new variable ; it then enters the equation
(1) merely as a parameter, in which form it also enters the
r\ *\ r\
operator x- — \-yz Mr- » when this latter is expressed in
r i>x ?>y cz r
the new variables.
We thus have an equation in two variables admitting an
operator, and can therefore find by a mere quadrature the other
integral.
CHAPTER VIII
INVARIANT THEORY OF GROUPS
§ 93. We have already defined transitive groups (§ 44), but
it is now convenient to give a second definition of such
groups, and to show that the two definitions are consistent.
The group
(1) X$ =Ji(pCi, •••> %n, d\, ...,a,r), [ir = 1, ...,7b)
is said to be transitive if amongst its operations one can be
found which transforms any arbitrarily assigned point into
some other point, also arbitrarily assigned.
The group will therefore be transitive if, and only if, the
equations (1) can be thrown into such a form, that some n of
the parameters ax , ..., ar can be expressed in terms of xx , ...,xn,
x[ , . . . , afn and the remaining parameters. The group cannot
then be transitive unless r ^ n. The group will be transitive
unless all ?i-rowed determinants vanish identicallv in the
matrix . , . ,
OCCl dCCx
<)a1 7>ar
Hi
IK
If we recall the rule for forming the infinitesimal operators
we shall see that the group is transitive unless every n of
those operators are connected ; and we thus see that the two
definitions are consistent.
The group is transitive therefore if, and only if, it contains
n unconnected operators. If r = n the group, if transitive at
all, is simply transitive ; and in this case there are only a
discrete number of operations which transform an arbitrarily
assigned point into another arbitrarily assigned point.
The mere fact that r ^ n is not enough to secure the
CAMPBELL
114 ONLY INTRANSITIVE GROUPS [93
transitivity of the group ; thus we saw that r was equal to n
for the group of rotations about the origin, viz.
a d a d a a
^ 2>z i>y <^x 2>z dy ° dx
but the group is not transitive, for these operators are
connected.
An intransitive group cannot therefore have n unconnected
operators. Let such a group have q unconnected operators ;
we shall now prove that these form a complete system.
Let Xlt ..., X be any q unconnected operators of the group,
and let the other operators be X +1, ...,Xr then
where <f>„+; &, ... are known functions of xlt ...,xn.
We have
(Xi*Xk) = 2<CiksXs — 2(Ctfe8 + 2^Ci,k,q+j(t)q+j,s)Xs>
where i and k may have any values from 1 to q, and therefore
Xl, ..., X form a complete system.
If a function is annihilated by these q operators Xlt ... , Xq,
it must also be annihilated by X +1, ... Xr; and therefore on
considering the canonical form of the group we see that such
a function is unaltered by any transformation of the group.
We have proved that there are (n—q) functions annihilated
by X15 ...,X , and we therefore conclude that an intransitive
group has (n — q) unconnected invariants.
§ 94. To express this result geometrically we look on
x1) ...,xn as the coordinates of a point in 7i-way space, then
Jl\X1,...,Xn) = Ctj, . .., Jn-q \Xi, •••j^'jj = ^n—q
will be a g-way locus in this space, and the coordinates of this
locus are the constants ax, ...,an_q. We keep the form of the
functions /ls ...,fn_q fixed, but vary the constants, and thus
have these q-w&j loci (or g-folds) passing through every point
of space. If we take f1} ...,/w_„ to be the invariants of the
intransitive groups, then by the operations of the group
a point lying on one of these loci is moved to some other
point on that locus ; we say therefore that this decomposition
of space, into aon-Q g-folds, is invariant under all the opera-
tions of the group. Thus for the group of rotations about the
95] HAVE INVARIANTS 115
origin, space is decomposed into a simple infinity of spheres,
whose centre is the origin, and a point lying on any one of
these spheres can only be transformed to some other point on
the same sphere.
§ 95. Only intransitive groups can strictly be said to have
invariants, and the problem of finding these invariants is
equivalent to that of finding the integrals of the complete
equation system formed by their unconnected operators ; yet
we shall see that in several ways the idea of invariants can
be extended to transitive groups also. Two points of space,
#u •••»#» and yi,...,yn, which are transformed to two other
points by the same transformation scheme, are said to be
transformed cogrediently ; thus if
Vi = Ji (2/l5 •••' 2/?* s aiJ---jar/>
we should say that x1,...,xn and yx, ...,yn were transformed
cogrediently.
No function of the coordinates of a point is invariant for
the operations of a transitive group, yet there may be functions
of the coordinates of a pair of points, which are invariant
when the points are transformed cogrediently by the opera-
tions of a transitive group ; thus the transitive group
^x ~by ~bz
has the three invariants xx — x2, y1—y2, Z\ — z-n where x1} yx, zx
and x2, y2, z2 are two points cogrediently transformed by this
translation group.
We could say in this case that we have extended the point
SrouP 1 ± ±
Tix ^y 2>z
into the point-pair group
} ^ S S a d
dxx ox2 oyx dy2 ozx oz2
and this extended group is intransitive, and has the three
unconnected invariants xx — x2, y^ — y^ z\~z-2-
Similarly the group of movements of a rigid body, viz.
a
a
a
2) d
a d
a a
~i)X
ty'
te'
vYz~z^
<)x <)z '
x- 7/ —
<>y J <>x
I 2
116
GROUPS IN COGREDIENT
[95
is transitive and has no invariant ; yet when extended so as
to give the point-pair group
^>xx ^x2
2/1 ^ Zl*yi
d d d
■»;*.,
ty2
this group is intransitive, and has the invariant
(xx-x2f + {yx - y2)2 + (z, - z.2f.
This expression is therefore an invariant of the coordinates
of a point-pair, when cogrediently transformed by the opera-
tions of the transitive group of movements of a rigid body.
The reason why this extended group of six operators in
six variables has an invariant is that the operators are con-
nected, as we prove by considering the determinant
1 ,
o,
o,
1 ,
o,
0
o,
1 ,
o,
o,
1 ,
0
o,
o,
1 ,
o,
o,
1
o,
-%>
2/i,
o ,
-z,,
2/2
Zn
o,
#X5
^2'
o,
— ^2
Vu
iCl5
o,
-2/2>
x2,
0
and subtracting the first column from the fourth, the second
from the fifth, and the third from the sixth, when it is seen to
be zero.
Since five of the operators are unconnected there is no
other unconnected invariant of a point-pair for the operations
of the group of movements.
If we were to extend this group so as to apply to triplets
of points we should not get any really new invariants ; it is
only when the operators are taken so as to apply to point-
pairs that the six operators are connected; in the case of
point-triplets we should have six unconnected operators in
nine variables ; and therefore only three invariants, viz. the
expressions for the mutual distances of these points.
§ 96. The operators of the linear group of the plane, viz.
af=l1x + 'm1y, y' - l2 x + m2 y,
are
x
(>X
x— 5 y^-> y^—\
<>y J dx ^cty
and as two of these are unconnected the group has no
invariant.
96] SETS OF VARIABLES 117
If, however, a0xP + pa^'1 y + ...
is any binary quantic, the quantic becomes, on applying the
transformations of the group,
a'0x'p +pa[x,p-'i y' + . . . ;
and we often speak of those functions of the coefficients
a0, a1} ..., which are such that
as invariants of the linear group.
These functions are however invariants, not of the linear
group
af^x + m^y, y/=l2x + m2y,
but of the group
a'0 = (IqI-F + ..., a[ = aQl^-1ml + . . ., a2 = a0Z11'~2m12 + . . .,
of which the linear operators are Alf A2, A3, A±, where
^=^o4+(^~l)al4+(^"2)a24+•••+^-l^,
A2 = PaiT^ + (P- *) «2vT + W~ 2) a3 T7T + ••• +a;
p-
7) ?> 2> d
A,= a,— + 2«2r— + 3a3- |-...+_pa s
If we denote the operators x— by Xx, x— by X2, 2/ —
by X3, and y — by X4, we see that
X1 — A1, X2 — A2, Xz — A3, X4 — A4
are four operators, each of which annihilates the quantic
a0xP+pa1xP-1y+ ... ;
and that there is no operator of the form
a d d
<>a0 loa± oa2
118 INVARIANT THEORY OF [96
(where a0, a15 ... are functions of the coefficients a0, a^ ...
only) which will annihilate this quantic.
We must now express the invariant theory of binary
quantics in such a form as to suggest the extension to general
group theory.
First we verify the group property of Xl5 X2, X3, X4 by
noticing that
(X15 X2) = X2, (Xls X3) = — X3, (A15 XJ = 0,
(^2> ^3) = ^l — ^4> (^2» ^4) = ^2> 0^3 > ^4) = — ^3'
Next we see that the operator
annihilates the quantic, since each operator
X1 — A1, X2 — A2, X3 — A3, X4 — A4
annihilates it.
Since X15...,X4 are each commutative with At, ...,^.4
(being operators in different sets of variables), and since by
the group property
7i = 4
(Xi,Xh)-^cikhXh=0,
we conclude that
must annihilate the quantic.
Now this is a linear operator, not containing x or y ; it can
therefore only annihilate the quantic if the coefficients of
in it are identically zero : we conclude that
• • •
da0 ~da
y fc = 4
(-Ait -Ak) = ^cikh(-Ah);
that is, the operators — A13 —A2, —A3, —A4 generate a
group, and this group has the same structure constants as
the group Xlt X2, X3, X4.
§ 97. We shall now take X to denote the linear operator
e1X1 + e2X2 + e3X3 + e4X4,
and A to denote the linear operator
e1A1 + e2A2 + e2A3 + eiAi,
98] BINARY QUANTICS 119
where elt e2, e3, e4 are parameters unconnected with the
coefficients or variables in the binary quantic.
Since X — A annihilates the quantic we have
a0 xp +pa1 xp-1 y + ...= ejr~A (a0 xP+pa^ xp-1 y + ...),
= e~Aex (a0xP + pa1xp~1y + ...),
any operator Xi being commutative with any operator Aj .
The linear transformation
(1) x'— e*x, y'= eYy
gives ex(aQxP+pa1xp~ly+ ...) = a0x'p+pa1x'p-1y'+ ... ;
and therefore, since
a0 xp + pax xp-x y + . . . = a'Q x'p + pa[ x,p-1y'+...,
we conclude that
e~A (a0 x'p +pax x'p~l y*+ ...) = a'Q x'p +pa\ x,p-ly' +....
Equating coefficients of like powers of the variables on each
side, we see that
'.— c-A,
(2) 4 = 6-* a
i •
and so generally <p (a'0, a[, ...) = e A<f>(a0, alf ...).
It now follows from (1) and (2) that if
/ \xi y> ao> aii • • • » ap)
is any function whatever of x, y, a0i a1} ...
f{x', y\ a'0, a'x, ..., op = ex~Af(x, y, aQ, alt ..., ap).
§ 98. Covariants and Invariants, as defined in the Algebra
of Quantics, are therefore merely the functions annihilated by
Ax — Alt ..., A4 — Ai}
four operators which are unconnected, and which generate
a finite continuous group.
If we are given a group Xls ..., Xr and want to find the
invariant theory which will bear the same relation to this
group as the invariant theory of the Algebra of Quantics
bears to the linear group, we must find some function
(p {X^, ..., xn, Cj, ...,cm),
where c15 ..., cm are constants, such that for any transforma-
tion of the group we may have the fundamental identity
<p {x1} ..., xni Cj, . .., cm) = (p (x1} ..., xn, Cj, ...,cm;,
120
INVARIANTS ASSOCIATED
[98
cfx, ..., c'm being constants, which are functions of c1} ..., cm and
the parameters ax, ..., ar of the given group.
Following the analogy of the procedure in the theory of
binary quantics we should only take such a function as
satisfied no equation of the form
y^>r +-~+ym—- = °>
dcx
3c
m
where y15 ..., yTO are functions of c15 ..., cm only.
If the function found did satisfy such an equation we could
(since in it the parameters would not occur effectively) replace
it by a function containing fewer parameters.
Suppose now that we have found a function, with m effective
parameters, satisfying the fundamental identity
<p {X}, .,., Xn, C15...,Cm) — <p (iCj, ..., Xn, Cjj
Applying the identical transformation
cm)-
•» ^m
we have for it
(i= 1, ...,%),
j •"«>
cl> •••» cm/ — *r v^l> •••' ^»» cl» •••! cm/ »
and therefore, since x1, ...,xn are unconnected,
4=c&> (& = 1, ...,m).
We next apply the infinitesimal transformation
a&=
i = 1, ..., W'
and we must have, since c'h is a function of cx, ...,cm and
differs infinitesimally from c^,
c'k = ck + hhk (ci> • »i cm)> (jj. ~ j ' ' rm )
where yh]c, ... are functions of c15 ..., cm.
If then we denote by Ck the operator
m-
we see that
m
-^i + C^u •••} Xr + Cr
will each annihilate <t>{xx, ...,xn, cl,...,cm).
99] WITH EVERY GROUP 121
Proceeding as in the theory of binary quantics the operator
h = r
is seen to annihilate this function. Since no operator in
Cj, ..., cm only can do this, and since Xx, ..., Xr are commuta-
tive with Clt ..., Cr, we conclude that
h — r
and therefore Cx, ...,Cr generate a group with the same struc-
ture constants as the group Xls ...,Xr.
We do not, however, know that the operators C\, ..., Cr will
be independent ; and therefore the group which they generate
may be of an order less than r.
oince y i n v i n
generate a group, all of whose operators annihilate
(p yxx , . . . , xn , Cj , . . . , Cm)i
this group must be intransitive.
§99. When we are given the group X1,...,Xr we can
construct many functions of xx, ..., xn and a set of parameters
Cj, ...,cm, which will have the fundamental property of pos-
sessing an invariant theory ; it will be sufficient to show how
one such function may be obtained.
Let Alt ..., Ar, operators in the variables ax, ..., ar, be the
parameter group of X15 ..., Xr ; and let B1, ..., Br be the same
parameter group, but written in the variables bXi ..., br instead
of a15 ..., ar ; then
(1) X1 + A1 + B1,...,Xr + Ar + Br
is a group with r unconnected operators. This group must
therefore have {n + r) unconnected invariants, for it is a group
of order r in (n + 2r) variables.
If some one of these invariants does not involve xlf ..., xn
it must be an invariant of the operators
Ax + x>15 ..., Ar + Br ;
and as there are r invariants of this group, we see that there
must be n invariants of (1) which will be unconnected func-
tions of xlt ..., xn, but may also involve the parameters
122 INVARIANTS ASSOCIATED [99
aly...,ar, bx,...,br in addition to the variables x1,...,xn;
and some one at least of these invariants must do so ; else
would Xlt ..., Xr annihilate each of the variables xlt ..., xn
which is of course impossible.
We thus see that for any group there must always be a
function with the fundamental property
(") <r\xi> •••!''«) ci> •••scw<) — "M'''i5 •••) xn> ci' •'•icm)>
and therefore an invariant theory for each group.
The reason why we take the operators
X^A^B^ ..., Xr + Ar + Br
rather than the operators
X1 + Alf ..., Xr + Ar,
is that for the latter set of operators there can be no invariant
theory ; since, A1, ..., Ar being a transitive group, there are no
functions of al5 ..., ar annihilated by these operators.
We now take X and C to denote the respective operators
e1X1+ ...+erXr and e1C1+ ... + er Cr ;
and, as in the corresponding theory for binary quantics, we
have, since X + C annihilates <f>(x1} ..., xn, cl5 ..., cm),
9 0*19 •••s xn> C19 '"j cm) = e 9\aiJ •••» xn> cl> •'•>cmn
= e e 9 \pi *»! c\ s • • • 9 cm) »
and therefore
9 (^19 •••» xn, cx, ..., cm) = e (f> {xlf ..., xn, clt ..,, cm).
Since the parameters Cj, ..., cm enter the fundamental
function p effectively, we now have
c'i = ecCi, (i=l ro);
and more generally, if /(a^, ..., xni cx, ..., cj is any function
whatever, we must have
/ (Aj, ..., #n, c1} ..., cwJ = e^ J \xi> ••') xn-> ^19 •••> cm)*
The co variants are therefore those functions of x1,...,xni
cx, ..., cm which are annihilated by
X1 + C1, ..., Xr + Cr ;
and the invariants are those functions of cls ..., cm which are
annihilated by pp.
and therefore for every group we have a corresponding
invariant theory.
100] WITH EVERY GROUP 123
§100. For a given group Xv ...,Xr we may be able to
obtain a fundamental function without having to go through
the process of finding Glt . . ., Cr , and then finding the invariants
of Xx + C1 , . . . , X r + Gr .
Thus if we take the group of order ten Xly ..., X10, where
5 das dz 6 dy * da
_ d d d
' ox oy oz
d d d
: X8 = (tf + z*-x*)Tx -2*2/- -2«f^,
X9 = {z* + a?-tf)±-2xy^-2yz±,
X10=(x2 + f-z*)±-2yz±-2xz±,
ia group which transforms minimum curves into minimum
curves, we see that by any operation of this group the
{function
a1(x2 + y2 + z2) + 2g1x+2f1y + 2h1z + dl
{ } a2(x2 + y2 + z2) + 2g2x + 2f2y + 2h2z + d2
is transformed into a function of like form, but with a different
set of constants.
The function (1) being fundamental, the group in the para-
meters is Cz, ..., C10 where
°2 -~aicj1~2hoir1~a2cj2~Zf2od2
CO d d
c«=-^dT1 + ^-^d5 + ^^'
124
SPECIAL POINTS OF
[100
2)
¥2
vh^h.
+ 2d1^2d
a
2 ado
a _ a a 7 a
^1 tyl <>«2 *0S
^a1 d^x 2da,
— 2&g-t c?0 ^r-
c*/i.,
It may be verified that this group has the same structure as
A j , . . . , A 10 . ^ .-it
This group, though of the tenth order and in ten variables,
is intransitive, and has the absolute invariant
(2g1 g2 + 2/t/g + 2 hx h, -d^a2- a.2d^
(9i +fi + V-Mi) (&2 +/22 + h22-a2d2) '
Since the group Xls ..., X10 transforms spheres into spheres,
and surfaces intersecting at any angle into surfaces intersecting
at the same angle, we could have foreseen that the group must
have this invariant, for it is a function of the angle at which
the two spheres,
a1(x2 + y2 + z2) + 2g1 x + 2f1y + 2h1z + d1 = 0,
a2(x2+y2 + z2) + 2g2x+2f2y + 2h2z + d2 = 0,
intersect.
§ 101. We know that only intransitive groups can properly
be said to have invariant functions, but groups, whether
transitive or intransitive, may have invariant equations.
Before we consider the theory of the invariant equations
admitting a given group, we must prove the theorem quoted
in § 91 as to the transformations which a complete equation
system can admit.
Let F1? ..., Yq be the operators of a complete system where
a a
7* = r^a^
+ ...+vi»
i>X
(& = 1, ...,g),
u
101]
A COMPLETE SYSTEM
125
and let Y\, ..., F' be the corresponding operators obtained by
replacing xi by x\ in Yx , . . . , Yq , where
xfi = <pi{xv...,xn), (i = l,.,.,n),
is any transformation scheme.
We know that the equation system admits this transforma-
tion if, and only if,
(!) Y'k=iPkiYi+-+Pk<lYq> (k=l,...,q),
where pki, ... are functions of xli...txn such that the deter-
minant
Pin
P2n
Pig
Pqq
does not vanish.
Let r)1k denote the result of substituting x\, . . . , x°n for x1 , . . . , xn
respectively in r)ik ; and let the operator
*55i +
+4
2>
n*x.
n
., n),
be denoted by F£.
If «i = **(«{, ...,«!). (>=1.
we shall denote by F£ the operator
Suppose now that x\, ...,x°n is a point of order A, so that
not all /i-rowed determinants vanish in the matrix
tfi.
&.
71°
'hn
Vim
then exactly h of the operators Yf, ..., F£ are unconnected,
viz. F", . . ., F^ ; what we have to prove is in effect that h of the
operators Fx0, . . . , F° will be unconnected.
We have
where the functions o-^+^j., ... are functions of ccj, ..., cc° such
126
SPECIAL POINTS OF
[101
that none of them are infinite ; we also suppose that in the
neighbourhood of this point all the functions rji]c, ... are
regular ; that is, we assume that rjik = rf-^ + a, series of powers
and products of {xx — x\), ...,(xn — #°), and that in this neigh-
bourhood the functions p^, ... are regular and their deter-
minant does not vanish; and finally we assume that the
transformation / _ . / \
•*■» — fi v^i? •••j **-n/
is regular in this neighbourhood, so that rf^, ..
regular in the neighbourhood of aj, ..., xQn.
We now have
are also
r* = *?+&5Er + -+S
**1 **"**.
(k= 1, ...,q),
where the functions £fe-, ... vanish for xt = xf ; and therefore
l = q
j=h
t = h—q
Yk = 2 Puj Yj = 2 (P% + 2 Pa°. k+t <+t. ,-) 7}
+&i^+-+6»555
n
vanish for xi = x\.
where the functions (j.-,
We can therefore, if we take any (h + 1 ) of these operators
Ylt ..., Yq, say Y'lt ..., Y'h+1, find functions 0J, ..., 6%+1 of
., x°n such that
elY'1+...+di+1rh+1 = £ *+...+&
1>
'X,
<>z„
where £19 ..., £n vanish for o^ = a;? ; and therefore
is a function of x'x,...,x'n, x\,...,x°n which vanishes when
xi — x\\ and therefore, since xi = x\, if xi = x°{ , it vanishes
for x'i = x\.
We have thus proved that any (h+ 1) of the operators
Ff, ..., F", are connected, for we have proved that all
(A-fl)-rowed determinants vanish in the matrix
A,
77°
7/ln
/
'</«
Suppose now that only (h—s) of the operators Ff, ..., Y° are
102]
A COMPLETE SYSTEM
127
unconnected ; then just as, from the fact that exactly h of the
operators F", ..., F° were unconnected, we proved that any
(h+1) of the operators Fx°, ..., F° were connected, so we
could now prove that (h — s+1) of the operators Y°, ..., F° are
connected, and therefore s cannot exceed zero, so that exactly
h of the operators Ff, ..., F° are connected.
We have thus proved the theorem that, by any transforma-
tion which a complete sy stern admits, a point of any assigned
order is transformed to a point of the same order, provided
that the transformation is regular in the neighbourhood of
the point.
§ 102. We now take Xlt ..., Xr to be the operators of a
group where
X^= ^^+---+^n^r' (k=l,...,r),
and we say, as in the theory of complete systems, that a point
is of order h, if when we substitute its coordinates in the
matrix >. >.
£n> • • • Ci?j
6
rn
£
r n
all (7i+l)-rowed determinants, but not all A-rowed deter-
minants of this matrix, vanish.
We shall prove later that for any transformation of the
we shall have
X'j = ejiXi+ "' +ejrXr> 0' = !> •••> r)>
where e-&, ... are constants whose determinant does not vanish.
If then x\,...,xQn is a point of order h all the functions
£i]{, ... are regular in its neighbourhood; and, since now no
exceptional case can arise through a want of regularity in
any of the coefficients, we see, as in the case of the complete
system, that by any transformation of the group a point of
order h is transformed to a point of order h.
A point of general position is a point of order q ; there
are ccn of such points, for all (5f+l)-rowed determinants of
the matrix vanish identically, where q is the number of uncon-
nected operators ; if the group is transitive q = n. As there
128 INVARIANT EQUATION SYSTEM [102
may be no values of xx, ..., xn which make all ^-rowed deter-
minants vanish, there may be no special points in connexion
with an assigned group; if there are such points, there may
be a discrete number of them or there may be an infinity of
them ; if only a discrete number these points must clearly be
fixed points, unaltered by any operation of the given group.
Suppose that cc8 points will make all (A+l)-rowed deter-
minants of the matrix vanish, but not all A-rowed determinants
vanish ; and let
be the equations which define these points ; the theorem
which we have proved asserts that points satisfying these
equations will be transformed to other points satisfying the
same equations ; in other words the equations (1) admit the
operations of the group Xx, ..., Xr; that is, these equations
are invariant equations.
§ 103. Let
0) fl-«+m = rs+mri' ■")*«)) (Wl = 1, ..., ?l — s)
be any equation system admitting a group X1,..., Xr; we
shall now define a set of operators closely connected with the
system.
If f(xXi..., xn) is any function of xx, ..., xni we shall denote
by / the function f(xv ...,■ xgi <f>8+1) ..., <p7l) of the variables
xx, ..., xs ; and by Xx, ..., Xr the r operators
J. s
we call Xx, ..., Xr the contracted operators of Xx, ..., Xr with
respect to the equation system (1).
From the definition of the bar
m = n—s
- 111 = n — a - - -
and therefore
Xk.f={Xhxx)(^) + ...
m = n — x
+(***,,)(^)+~2(**«w)(^)i
104] AND CONTRACTED OPERATORS 129
but we also know that
(Xkf) = (Xkx1)(^I) + ...
: — : — m = n — s
+<*>*}&) +2 KK3(s^-).
so that
TO= >l — S
(2) <x„/) = jr.7+2 <**(».+«- ♦.♦J) (e^-)-
Now the equations (1) admit the group, and therefore in
particular admit the infinitesimal transformations, so that we
must have
(xh{xs+m-4>s+m)) = 0;
and therefore from (2)
(Xhf)=Xh.f, (k=l,...,r);
that is, the result of first operating with Xk on any function of
the variables, and then deducing the corresponding function
with the bar, is the same as that of first obtaining the function
/, and then operating with the contracted operator Xh ,
§ 104. We can now prove that Xlt ..., Xr generate a group.
From the second fundamental theorem
(X{, Xj) =2^cijh Xk,
k = r
and therefore X{ £jm - X- £im = 2 %'h £km 5
consequently we must have
k = r
i £jm / €im ^ Cijk £&m'
and therefore from what we have just proved
h = r
i ' \jm ,/ • £im —- cij k ^km >
k = r
that is, (X{, Xj) = 2 Cijk xk.
It is not, however, necessarily true that the r contracted
operators will be independent.
CAMPBELL J£
130 EQUATION SYSTEMS [104
If the equations
0) x,+m = <t>8+m(xi>~ "xe)> (m = 1, ..., ?i-s)
are taken to be the equations which define points of order h
with respect to the group, Xv...,Xr, we know that these
equations will be invariant under the operations of the group ;
we shall now prove that h of the operators Xt, ...,Xr are
unconnected.
From the definition of a special point of order h, exactly h
of the operators
» . Ti
fuc +'~ + hi
(iX
(k = l,...,r)
'n
are unconnected ; and therefore not more than h of the operators
X1, ..., Xr can be unconnected.
Also since the equations (1) admit the group Xx, ...,Xr
and from these equations it follows that not less than h of the
operators X1, ...,Xr can be unconnected; we therefore con-
clude that exactly h of these operators are unconnected.
§ 105. We are now in a position to determine all the equa-
tion systems admitting a given group.
If the system of equations
(1)
x
s+m
— $s+m(A'l> •••»#«)> vn ~ 1> •••j'^' — S)
is to admit all the transformations, it must in particular admit
all the infinitesimal transformations of the group, and there-
fore we must have
iXjxs + m) = (Xj4>s+m(Xl> ■••>««)), (m = j' \'^n_s)'
Conversely, if the system admits all the infinitesimal trans-
formations, it will admit all the finite transformations of the
group; for let f(x1, ..., xn) be any function of the variables,
then we have proved that X1,...,Xr being the contracted
operators of Xy, ..., Xr with respect to the equations (1)
Xj=X:.f,
105] ADMITTING A GROUP 131
and therefore
(e1X1+...+erXr)f = (e1X1 + ...+erXr)f,
and (e1 Xx + . . . + er Xrff = (e1X1+...+er Xrff, and so on ;
if then/ is any function such that
X1f=0,...,Xrf=0,
that is, an equation admitting the infinitesimal transformations
will admit all the finite transformations of the group.
Suppose now that we are seeking an equation system
admitting a given group, the points, whose coordinates satisfy
these equations, must either be points of general position with
regard to the group or points of special position. Suppose
that they are points of order h, and that q is the number of
unconnected operators in the group X1, ..., Xr; if A. is less
than q the points are ones of special position ; if h is equal to
q they are points of general position, and h cannot be greater
than q (§ 91). We say that the equation system is of order h.
We now take
{*) xs+m = 4>s+m\xi> •••» xs)> ytn ■= I, ...,Tb — S)
to be the known equations giving the loci of points of order h ;
and XXi ...,Xr to be the known contracted operators of the
group with respect to these equations; and we take Xx, ...,Xh
to be the h unconnected operators of the contracted group.
Any equation system of order h must therefore by means of
the equations (1) be reducible to an equation system in the
variables a^, ..., xs\ and in order to find such a system it
is only necessary to find the equation systems admitting
Xt, ..., Xr. This equation system being of order h cannot
allow the points satisfying it to be special points with regard
to the group Xlt ...,Xr ; for were they so, they would be of
order less than h, which is contrary to our supposition.
The problem is therefore reduced to this ; we are given h
unconnected operators Xx, ..., Xh forming a complete system ;
and we have to find all the equation systems which admit
these operators, and are yet such that the points satisfying
these equations are not of special position with respect to
K 2
132
EQUATION SYSTEMS
[106
§ 106. By a change of the variables we can take Xlt ..., Xh
to be respectively
)a'-,
** = «
M
i
ox\
h
where &|, ... are functions of xv...,xh, and (s — A) other
variables which occur as parameters ; and the equation system
we are seeking must not make the determinant
Ml!
»7u»
£
i/i
•M
zero.
Suppose that/ (a?!, ...,#g) = 0 is one equation of the system
admitted, then
2>/ */
fen "5wT" + • • • + ti7(
t>#j
i>#7
and therefore, since the determinant is not zero, we must have
>/_
*f
^x1~0,'"'^x
= 0.
//
The required equation system can then be only a system of
equations in the variables xh+1, ...,xa; that is, the system of
equations can only connect the common integrals of
r1(/) = o,...,^(/)=o.
Example. Consider the group of the fourth order,
V z —
U oz oy
i
ex
o o oooo
x — , x- « — j x-- + yK — \-z^r-
oz oy * ox ox ey oz
This group is transitive, and its matrix is
0, -z,
z, 0,
-y, »,
x, y,
y
■X
0
z
rei
n
106] ADMITTING A GROUP 133
The only values of x, y, z which cause the determinants of
the second or lower orders to vanish are x = y = z = 0 ; and
obviously there cannot be contracted operators to correspond
to a discrete number of special points.
Forming the determinants of the third order, we see that
the equation x2 + y2 + z2 = 0 causes all of these determinants
to vanish ; this equation is therefore admitted by the group,
and defines points of order two. The contracted operators
with respect to this equation will therefore form a group in
two variables, and will have two unconnected operators, and
cannot therefore have any common invariants, so that the
only equation admitted by the group is the equation
x2 + y2 + z2 = 0.
Example. Consider the simply transitive group
. „ „ «v O <) <)
(ir + z*— xr)- 2xy- 2xz — *
Xif 'lx b dy Zz
. _ _ ov o h f)
(or + sr — y£)- 2xy- 2yz~->
v ty <>% u 2>z
. „ _ _. o d d
(x2 + y2—z2)^ 2xz-r 2yz — >
v a '^z Ix J ty
The matrix is seen to be (x2 + y2 + s2)3, and when we equate
this to zero we see that all determinants of the second order
vanish, so that the equation
z = i(x2 + y2j*
(where the symbol i denotes */ — 1) defines the locus of
points of order one. This is the only invariant surface with
respect to the group; to obtain the invariant curves with
respect to the group we must find the integrals of
dx ° ^y
since the contracted operator is
xTx+y-^-
The invariant curves are therefore
y = ax, x2 + y2 + z2 = 0,
where a is a variable parameter.
134 EXAMPLE [106
It must not be supposed that an invariant of the contracted
operators is an invariant of the group itself; in transitive
groups they never could be such : in this example - is an
invariant of the contracted operator, but for the given group
it is only invariant on the surface x2 + y2 + z2 = 0.
If we take the group of order ten which transforms minimum
curves into minimum curves, we see that since it contains
^- j r— ? r— one of the determinants of its matrix is unity,
dx <>y cz * J
and therefore there are no special points with respect to this
group ; and because it is transitive, and without special points,
it cannot have any invariant equation.
CHAPTER IX
PRIMITIVE AND STATIONARY GROUPS
§ 107. We have seen that for the group which transforms
minimum curves into minimum curves there is no invariant
surface, but, since it transforms the sphere
a(x2 + y2 + z2) + 2gx+2fy + 2hz + d = 0
into some other sphere, it has an invariant family of surfaces,
viz. the spheres in three-dimensional space.
The theory explained in § 99 would show us that for any
group whatever we could find invariant families of surfaces.
One case of this general theory is of particular interest, viz.
when the number of parameters in the surface is less than
the number of variables. Following the usual phraseology,
we shall call the parameters involved in the equation of any
surface the coordinates of the surface.
When the number of the coordinates of a surface is less
than the number of variables we may express its equations
in the form
c15 ..., cn_q will then be the coordinates of the surface; and,
since a point on it has q degrees of freedom in its motion, we
say that the surface is a g-way locus in ^-dimensional space,
or briefly a q-iold.
We suppose the forms of the functions 019 ..., </>w_2 to be
fixed; if for all values of the coordinates c1,...,cn_q of the
q-iold, the g-fold admits the transformations of the group
JT19 ..., Xr the group must be intransitive. Since the <?-folds
can only each individually admit the group when <f)x, ..., </>w_2
are invariants of the group, we see that the group cannot
have more than q unconnected operators.
Suppose now that the group is intransitive, and that
xq+i> '"> xn are iks invariants ; we then have
^k = €kl\Xl> '"> X7i) >^T" + ••• + €kq V*l' "'J xn) jTJT-' \^= *> •••>rr
136
PRIMITIVE AND
[107
The equations xq+1 = aq+1, ..., xn = an are invariant for the
group ; suppose that .i\, ..., xq, aq+l, ..., an is a point of general
position, the contracted operators with respect to these
equations are Xv ..., Xr, where
d
A j. = fji (Xv ..., Xqi Uq + i, •••) dn) >T— + • ••
"1" C A;^ V^l ' • • • ' xq ' a2 + l> •••> &7J ^
a?,-
We know that these contracted operators will generate
a group, and that q of its operators will be unconnected, so
that this group, being in q variables, will be transitive.
If we say that the transformation
Q-i = eei^i+... + erJTrXi} ^ _ ^ ^ n^
in the group Xls ..., Xr corresponds to the transformation
Xi
ee1X1 + ...+erX
*X
n
(i = l,...,n)
in the group Xlt ...,Xr; then any point on the g-fold
Xq+1 = Clq+i, ..., Xn = an
is transformed to the same point on that g-fold by either of
these corresponding transformations.
Now the group Xlt ..., Xr is transitive, and therefore any
arbitrarily selected point on this g-fold can by the operations
of this group be transformed to any other arbitrarily selected
point on the q-fo\d : it follows that by the operations of the
group Xx, ..., Xr any point on this q-fold can be transformed
to any other point on the same q-iold.
§ 108. Without, however, assuming that any one of the
g-folds . . (
<Pl \X-j_, ..., Xn) — Cj, ..., <pn_q \XX, ..., Xn) — cn-q
is transformed into itself by the operations of the group, we
shall suppose that the totality of them is invariant ; that is,
the g-fold with the coordinates c1, ..., cn_q is transformed to
the g-fold with the coordinates cx, ..., c'n_q, the forms of the
functions </>1} ..., (pn_q which define the g-folds being of course
fixed.
If x1, ..., xn is a point on
<Pi\%i, ..., xn) — cv ..., <f>n_q (x, ..., xn) — c
n-q)
109] IMPRIMITIVE GROUPS 137
and if this point is transformed into x[, ,.., x'n then we must
nave , , f ■. r , / / .. /
9l \*d •••) ^jj — ^i> "•> Qn-q v**l> •••> *^n/ — ^M-g '
but unless the group is intransitive, and tf>v ..., <f>n_q are its
invariants, we cannot have
<Pi {xx, ..., xn) = 9>x (aj15 ..., #W/), ...,
If, however, the totality of g-folds is invariant we have,
whether the group is intransitive or not, an invariant decom-
position of space into con~« g-folds.
A group under which some decomposition of space is
invariant is said to be imprimitive ; a group under whose
operations no such decomposition is possible is said to be
primitive; thus intransitive groups are a particular class of
imprimitive groups, and primitive groups are a particular
class of transitive groups.
§ 109. Let
( / xi = Ji (**!» '"> ^n> ®1> •••' ®r)> (£ = 1, ..., TlJ
be the equations of the given group, and let
(p1{Xli ..., Xn) = Cj, ..., <Pn-q\%ii •••> %n) = cn-q
be an invariant decomposition of space ; when we apply to
this g'-fold the transformation (1) we get
and we must therefore have an equation system of the form
<4 = V^'Oii ••• >c»-2> a15...,ar), (i = 1, ..., w-g).
It follows therefore from our first notions of a group that
the functions \^15 ..., \^ra_„ will define a group containing the
identical transformation and r infinitesimal transformations,
though these are not necessarily independent.
The variables in this group are the coordinates of the g'-folds
in space xx, ..., xn, and we may say that we have passed to
a new space in (n—q) dimensions ; to any assigned point in
this new space there will correspond a definite q-fold in the
space xx, ..., xn; and to any transformation
^i = Ji V^l ) "'i^B! ai j • • • j ttr/' I* = ' • • • ' Tv
138 COMPLETE SYSTEMS AND [109
in the original space there will correspond a transformation
U = ^i(cl> •••» cn-q> «i>»-> ar\ (l=l,...,n~q)
in the new space.
By a change of the variables we may take
xq+l = cg+i» •••> xn = cn>
to be the equations of any g-fold, whose family is unaltered
by the operations of the imprimitive group X1, ..., Xr.
In this system of coordinates the finite equations of the
imprimitive group must be of the form
%i = Ji v^i5 •••> xn> ^1» •••' ^r/' \P = > • '•> 3/>
&q+j = Jq+j \%q+l> "•> %n> ai> •••> ar)> \J = 1 > . . . , 9X — q) ',
for any g-fold of the system must by the operations of this
group be transformed into some other.
The infinitesimal operators of the group are now
where £ftj g+j* ••• do n°t involve a^, ..., #9.
It therefore follows from the identity
h=r
(Xi> x-k) = 2 cihh %h
that the r operators Z1 , . . . , Zr> where
Zh=h,q+iTZ — +... + &„ C— » (A= l,...,r),
°^2+l oxn
form a group, such that
this group, however, is not necessarily of order r since the
operators may not be independent.
§ 110. The complete system of equations
5-^ = 0,..., r^-=0
da^ oxq
is invariant under all the operations of the imprimitive group
Xj, ..., Xr. This is at once seen to follow from the fact that
€k,q+j> -..do not involve xlt ..., xq.
Ill]
IMPRIMITIVE GROUPS
139
Conversely, if any complete system is invariant under the
operations of a group, that group must be imprimitive. For
by a change of coordinates we can take the complete system
to be
and then, if
J = 0 ■ J = 0
is an operator of the group which the system admits, we see
that £]c,q+ji'" cannot involve a^, ..., xq\ and therefore the
equations _ .
can only be transformed to equations of the form
xq+l == cq+l> "•> xn = cw
that is, the group is imprimitive.
§ 111. We have now seen that groups may be divided into
transitive and intransitive classes of groups; and also into
primitive and imprimitive classes ; there is yet a third
division into stationary and non-stationary groups. To ex-
plain this last division, let X1} ..., Xr be the r operators of the
group where
•^ 7c = £&1 \xl> • • •' xn) vTT + • • • + £fcn \xi j • • •? xn)
Q Jb-t
~hx,
n
(k= l,...,r),
and suppose that exactly q of these operators are unconnected,
say Xl}...,Xq
and let
(1) Xq+j=^<t>q+ji1c(x1,...,xn)Xh, (j=h...,r-q).
Let x\, ..., sc° be a point of general position, that is, a point
such that not all g-rowed determinants in the matrix
e
a'
e
In
£gl> • £qn
vanish, when the coordinates of this point are substituted in
it. First we see that any infinitesimal transformation of the
form
xfi = xi + t(e1X1 + ... + eqXq)xi, (i = 1, ...,n)
140 THE GROUP OF A POINT [111
will transform the point ajj, ..., a° to some neighbouring point;
for if the point remained fixed we should have
«i&+...+«2£$i= °. (*= *• ■'••!»). j
and therefore all g-rowed determinants of the matrix would
vanish.
The necessary and sufficient conditions that
ex X1 + . . . + er Xr
should not alter the point a§, ...,a£ are
%&+...+«,.&= o, (* = 1,. ..,*);
and these equations may by (1) be written in the form
k = q j = )•— q
2(e& + 2e2+i05+i,fc)^H = 0, (* = 1, ...,»).
Since then the point a^, ..., a° is one of general position, we
must have
j = r-q
and the general form of an operator of the group which does
not alter this point must be
j=r—q k = q
2 Vi (AVi ~ 2 <Vi, k K> • ••! xl) xh)-
It follows, since the transformations which leave a given
point at rest must obviously have the group property, that
the (r — q) independent operators
Jt = q
Xq+j-^<t>°q+j,JcXh> (j=l,...,r-q)
generate a sub-group.
We call this sub-group the group of the point scj, ...,a&!
Unless all the operators of a group are unconnected, to each
point of general position there will correspond one of these
sub-groups.
§ 112. Let now y\, ...,y°n be any other point of general
position, we now wish to see whether all those infinitesimal
transformations of the group which leave x\, ..., xfh at rest
have the property of also leaving $, ...,y°n at rest; that is,
whether the groups of the two points are the same.
113] STATIONARY GROUPS 141
If the groups of the two points are the same then for all
values of the parameters eq+li ...,er
j = r—q k=q
^ eq +j \Xq +j — ^ <t>q +j, Jc(Xl> •'•> Xn) ^k)
j = r—q k = q
= 2 Vi (xi+j ~ 2 4>q+j, h (y°i> — > 2&) Xh)>
where eq+1, ...,cr is some other set of parameters not involving
#2 , . . . , Xn .
Since the operators X1,...,Xr are independent, this can
only be true if . _ , . _ ,
and if further
j = r—q
2 eq+j (<Pq+j, k (A> -. <)-<Pq+j,k(fl> — M) = 0.
Now e„+1, ... er are independent, so that we must have
as the necessary and sufficient conditions that the groups of
the points teg, •..,#& and 2/i» •••> 2/« may coincide.
§113. The sub-group which leaves jkJ, ...,aj® at rest will
therefore leave at rest all points on the manifold
(!) ^q+j,^ (xl> •••' xn) = ^g+j.ftO*!' ■•• ^ra)'
,j = 1 r— q*
rj = ^ '— *i\
tjbssl,..., 2;
Of the functions ^q+j^'" not more tnan ™ can ^e un"
connected ; if n are unconnected only a discrete number of
points will lie on this manifold ; and we then say that the
group X1? . . ., Xr is non-stationary. If, however, fewer than n
of the functions are unconnected, say s, then the equations (1)
define an (n — s)-way locus; and the group of the point
x\, ..., x°n leaves invariant the continuous (n — s)-way locus
which passes through the point ; in this case we say that the
group Xx, ...,Xr is stationary. The groups of all points on
this locus are the same ; we shall call this locus the group
locus of any point on it.
If af1=fi(x1,...,xn,a1,...,ar), (i=l,...,n)
is any transformation of the group X19 ..., Xr, and X[, ..., Xr
142 STATIONARY GROUPS [113
are the operators obtained by replacing xi by a£ in Xx, ...,Xr,
we know from the discussion in § 75 that X[, .„, Xr are an
independent set of operators of the group X1,...Xr. Suppose
that by this transformation the point 03$, ...,a& becomes the
point x\, ...,x%; then,
e1X1+ ...+er Xr
being an operator which leaves x\, ..., x% at rest,
ex X'x + . . . + er X'r
will be an operator leaving x\,...,x% at rest; and the group
of the point x\, ...,a£ is therefore transformed into the group
of the point x\, ...,<• If then the group is stationary, the
(H_s).Way group locus through x\,...,x°n is transformed to
the (n — s)-way group locus through x\, ...,aPa. It follows
therefore that a stationary group is imprimitive, since the
group loci are transformed inter se.
It should be noticed that not all imprimitive groups, nor
even all intransitive groups, are stationary ; primitive groups
however, having no invariant decomposition of space, must be
non-stationary.
§ 114. We shall now give an analytical proof of the theorem
that the equations
(!) 4>q+j,k(xl>"->Xn) —cq+j,hi Ijfc p= 1 ..., q'
define an invariant decomposition of space into co* (n — s)-way
loci, where s is the number of the functions 4>q+j,k-> ••• which
are unconnected.
From the fundamental group property
(*,.jw -2w*«** (» = I; "'.;'"?! * = i--r)>
and from the identity
*=9
(2) IH=2*g+i.»J»» (j= 1, ...,r-<?),
we deduce that
»i = v, A- = 7 fc = g » = f
2 Gpfcm ^g+i, fe ^m + 2 ("^p Qq+j, k) %k = 2 Cp, q+j, i Xi-
If we apply to this the identity (2) so as to eliminate the
115] ARE IMPRIMITIVE 143
operators Xq+1, ...,Xr, we can equate the coefficients of
Xx, ..., Xq on each side of this identity, for Xx, ..., X are by
hypothesis unconnected ; we thus obtain
i = ?• — q k = q
P *2 +i> m = CP' 1 +J> m ~ CP> 2 +i> q+i^q +i, m — cp, k, m Vq +j, k
k — q, i = r — q
~ 2L cp,k,q+i<Pq+j,k<Pq+i,tn'
It therefore follows that by the infinitesimal transformation
x'i = xt + tXpx{, (i = 1, ..., n)
all the points which lie on any one of the (n — s)-way group
loci (1) are so transformed as to be points lying on some one
other of these loci.
We may perhaps see this more clearly if we throw (as we
may by a change of coordinates) the equations
(rq+j,k\xl> •••' xn) = Cq+j,k
into the forms
(3) %x = Cj, ..., xg = cs.
What we have then proved is that by any infinitesimal
operation of the group, and therefore by any finite operation
of the group, the coordinates x1 , ... xs are transformed into
functions of xx, ..., xs ; and therefore the (n — s)-way locus (3)
into the (n — s)-way locus
xx = yl5 . .., xs = ys
where yx, ..., yg are functions of cls ...,cs and the parameters
of the group X1,...,Xr.
§ 115. The functions <pjlx (xx, ..., xn) have only been defined
for the case j > q, fj. > q ; it is convenient to complete the
definition by saying that when these inequalities are not
satisfied fa^ (xx, ..., xn) is to be taken as identically zero.
We now define a set of functions Ui)lc, ... as follows:
t = r-q n = q n=zq,t = r-q
riyx- = C;ji.+ 2 CM,q+t <t><l + t,k + ^Cflik(t>JIX+2, C^i^+t <f>jlx <f>q+t,k-
If j > q,
t — r—q
Hijk — €ijk + 2d Ci,j,q+t ^q+t^k'
144 STRUCTURE FUNCTIONS [115
if k > q,
and if j > q and k > q,
^ijk = cijlr
Since c^h + CjiJc = 0 for all values of i,j, k we have
Xp^q+j,k = Up,q+j,k'
k — q
SinC6 Xq + i = ^<t>q + i,kXk>
k = q
Xq+i(t>q+j,m = 2*<$>q+i,kXk<$>q+j,m'
and therefore
*=a / i = 1, ..., r— <^
^g+«,2+J,m =-^ *rq + i,k '■*k,q+j,m> I ^ = lj««vr — <?
\m = 1, ..., gv
these are identities, satisfied by the functions <£r/+J- &, ....
Again, since
i' = r k — q t = r — 2
(^»J ^}) = 2 Cijk %k = 2 (C#ft + 2 ^,i, ry+i ^g + <, fc) -^fc .
we see that, X1} ..., X being the unconnected operators of the
group,
k = q ,
we therefore call the functions IT^j., when none of the integers
i, j, k exceed q, the structure functions of the complete system
The functions (frq+ik, ... we shall call the stationary func-
tions, since they determine whether the group to which they
refer is stationary or not.
116. Suppose that s of these stationary functions are uncon-
nected ; we can by a suitable choice of new variables bring
them to such a form that they will be functions of the
variables xlt ..., x8 only; and we can also express the
variables a^, ..., xg in terms of the stationary functions.
The equations
\l) xx = c1 , . . . , xs = cs
now give a decomposition of space which is invariant under
117] AND STATIONARY FUNCTIONS 145
the operations of the group Xr, ..., Xr ; only if s is less than
n can we say that the group is stationary ; and only if s is
less than n can we say that the equations give a decompo-
sition of space at all.
The operators of the group are Xls ..., Xr where Xk is
&i^r + --- + &w^r' (fc=l,. ..,*"),
1 74
and ffcl, ..., ifo are functions of a^, ..., x8 only; for the
(^—s^-way locus (1) must by any operation of the group be
transformed to some other (n — s)-way locus of the same
family. If therefore
Zlt ..., Zr will generate a group, such that
where the structure of the group X15 ..., Xr is given by
The group Z14 ..., Zr is not, however, necessarily of order r,
for its operators may not be independent.
We can construct this group Zx, ..., Zr merely from a knoiv-
ledge of the structure constants and the stationary functions
of the group Xlf ..., Xr.
For if the stationary functions are known it merely requires
an algebraic process to bring them to such a form that they
are functions of x1,...,xs only. We can then say that
xv ..., xs are known functions of the stationary functions;
and, since Xrf^ = Uitq+jih, and nitq+jfh is known in
terms of the stationary functions, we see that Xi <f>q+jtk is also
known in terms of them. It follows that X^x^ ..., XiXs are
all known functions, that is, the coefficients of r — » •••> =■ —
in X15 ..., Xr are all known; that is, the operators Z^ ..., Zr
are known when the structure constants and the stationary
functions are known.
§ 117. We have seen that the operators of an intransitive
group can be simplified when we know its invariants ; what
we are now about to show is how by a suitable choice of
CAMPBELL L
146 THE OPERATORS OF A [117
new variables to simplify these operators, and at the same
time to simplify the stationary functions <t>»+j, k (xv •••> xn)> ••• •
We so choose the variables that the stationary functions are
functions of the variables xlt ..., xs only.
Of the invariants of Xlt ..., Xq, the unconnected operators
of the group, some may be functions of x^, ..., xg only; if we
suppose that there are m such invariants, we may so choose
the variables that these are xl,...,xm; and m is not greater
than the lesser of the two integers n — q and s.
Since the stationary functions are now functions of xlt . . ., xs,
and x1, ..., xm are invariants of Xlt ..., X , we have
where £& m+1, ..., £&)S are functions of a^, ..., xs only.
Any function of xx, ..., #m is an invariant of Xlt ..., X„, but
there are (n — q — wi) other invariants, unconnected with these.
Let /(#!, ..., xn) be one of these other invariants; since by
hypothesis xli ..., xm are the only unconnected invariants
which are mere functions of xXi ..., x8,f cannot be connected
with xx, ..., xg ; we may therefore again so choose the variables
that/ will be xn.
In this system of variables the stationary functions are
still mere functions of xli ..., xs, and a?j5 ..., xm, xn are invari-
ants of the group.
There now remain (n — q — m—1) invariants, unconnected
with xlt ..., xm and xn; let f{xY, ..., xn) be one of these, we
next prove that it cannot be connected with xx, ..., xs, xn.
Suppose, if possible, that it is a mere function of Xj, ,..,x8,xn;
then, since it is annihilated by Xlt ..., Xq, we must have
3/ 3/
Sfc,ro+1 TZ + ••• + «« jTT — °> (* = !> •••><?)>
^TO + l °^S
for £&M = 0, because #w is by hypothesis an invariant.
Now ik m+1, ..., ^.s do not contain xn; and therefore, if an
is any arbitrary parameter,/^, ..., xs, an) will be annihilated
by Xls ,„, Z., As we have proved that no function of
aij, ...,#s can be so annihilated, unless it is a mere function
of xli...,xmi we conclude that f{x1, ..., xs, xn) is a func-
tion of ajj, ...,£rm and xn only; that is, it is not one of the
(n — q—wii— 1) other invariants. We can therefore by a fresh
choice of the variables take the function/ to be xn_1; and in
these new variables the stationary functions will still be
118] GROUP IN STANDARD FORM 147
mere functions of x1} ...,xs, and xlt ..., xm, xn, xn_x will be
invariants.
Proceeding thus, we see that we may finally take the
stationary functions to be functions of the variables xlt ...,x8
only, and may take the (n—q) unconnected invariants of the
group to be «15 ..., xm, xq+m+1, ..., xn.
In proving this we have implicitly proved the inequality
q + m ^ s.
When a group is brought to this form we say it is in
standard form.
§ 118. The above is the general method of bringing a group
into standard form when it is intransitive, stationary, and
when some one at least of the invariants of the group is
a function of the stationary functions ; the modification when
any one of these conditions is not satisfied is simple, and the
labour of bringing the group to standard form is lessened.
Thus, if the group is transitive, q = n, and m = 0 ; to bring
the group to standard form involves only the algebraic pro-
cesses of selecting the stationary functions in terms of which
the others can be expressed, and taking them as a new set of
variables x1} ..., xg.
If m = 0 then q ^ s, and the invariants may be taken to
be a„+l! ..., xn, while the structure functions will involve
xv ...,xs only.
If the group is non- stationary s = n and m = (n — q), and
the invariants are xlf ..., xn_q, while the structure functions
involve all the variables xx, ..., xn.
We saw in § 45 that in order to bring the equations of
a group, given by its operators X15 ..., Xr) to finite form it
was necessary to find the invariants of
e1X1 + ... + er Xr .
This problem is simplified for stationary groups ; for, when
we know the operators, we know the stationary functions,
and can by algebraic processes bring the above operator
to the form
k = r,j = s k = r, t = n—s
2 eh £kj («1> -.^^+2^ ijc,s+t (^ —i xn) ^— *
J s+t
There are (s — 1 ) unconnected invariants of this operator
which are functions of x1} ..., xs; and these may be found by
integration operations of order (s— 1) : having found these,
the remaining (n — s) invariants may be found by integration
operations of order (n — s).
L 2
CHAPTER X
CONDITION THAT TWO GROUPS MAY BE
SIMILAR. RECIPROCAL GROUPS
§119. The functions <i>q+j,k> '" which determine whether
a given group is stationary or non-stationary are of much
importance in other parts of group theory ; we shall now con-
sider their application to the problem of determining whether
two assigned groups are or are not similar ; that is, whether
or not the one group can be transformed into the other, by
a mere change of the variables.
Taking X15 ..., Xr to be the operators of a group of order r
and Ar1? ..., X to be the unconnected operators of the group,
we have
k => q
If we change to a new set of variables given by
Vi = fi \xl> •••» xn)> V' == 1 » • • • j ^)»
the r operators Xlt ..., Xr will be transformed into r inde-
pendent operators Y1, ..., Yr, where
7j; •, ... being functions of the variables ylt ...,yn.
At the same time the functions 4>„+j,k(xi> •••> xn)> ••• wu^ De
transformed into functions
r q +j, h \Vi > • • • j Vn)' • • • J
such that
lis — 1 ^ •••) Q
We must have
Jfc=r
(X^, Zy) =^cijkXk, and X,. = F^.
k = r
since
120] SIMILAR GROUPS 149
If then we have two groups, viz. Xls ..., Xr in the variables
xv ..., xn, and Ylt ..., Yr in the variables ylt ..., yn, each group
being of the rth order, we see that these groups cannot be
similar unless we can find a set of independent operators
Zx, ..., Zr, dependent on the operators Fl5 ..., Yr, and such
that the structure constants of Zla ..., Zr are the same as those
of the group Xly ...,Xr; and also such that Z1}...,Zq are
unconnected, and Z q+1, ..., Zr connected with Z1, ..., Z .
These conditions are necessary ; suppose that they are
fulfilled; we may then assume that the group Ylt ..., Yr can
be presented in such a form that the structure constants of
Yj, ..., Yr are the same as those of X15 ..., Xr, that F15 ..., Y
are unconnected, and that Yq+l, ..., Yr are given by
h = q
Yq+j =2vvi,fc(2/i' ->yn)Yk> U = h •••»*•-?)•
If the groups are to be similar we must further have
<i>q+j,k(Xl,~»Xn) = 'l'q+j,li(yi>-~>ynh ( 7, _ , „)'
If from these equations we could deduce an equation
between xli...ixn alone or between yly ...,yn alone, it is clear
that the groups could not be similar ; it will now be proved
that if no such relation can be deduced the groups are similar.
§ 120. Suppose that of these q(r — q) functions
exactly s are unconnected, we know that s>w ; between any
(s+ 1) of these functions there must be a functional equation ;
and therefore, since there is no equation connecting yt, ..., yn,
there must be the same functional equation between the
corresponding functions of ylt ..., yn.
It must be possible to find at least one transformation
scheme
y,i=fi(yi,>~>yn)> (i=i,...,n)
which will transform any s of the functions
Tq+j,k v2/i > ■••> 2/w/> •••
into the respective forms
rq+jjk '2/l' •••' 2/rz/' ••• 5
and therefore, since the same functional equation which con-
nects any (s+1) of the functions V'o+j, &> ••• "^^ connect the
150 CONDITION THAT TWO [120
corresponding (s+1) functions ^q+j,Je» •••> we 8ee that this
transformation scheme will transform each of the functions
^q+j,ii(y\> •••> 2/J> •••> into tne corresponding function
The theorem which is to be proved is therefore reduced to
the following : X1,...,Xr and F15 ..., Yr are two groups, each
of order r, in the variables 05j , . . ., xn and yx, ...,yn respectively ;
the operators in the first group XX,...,X„ are unconnected,
and
Xq+j =^<t>q+j,k(xl, <~>Xn) xk> U= 1,. ..,«"-});
in the second group F15 ..., Fg are unconnected, and
these groups will be similar if
h = r
(X{, Xj) = 2C#& -^ft»
and ( Y{ ,Yj) =2 c^* ^^ .
If by the transformation scheme
the stationary functions of X1? ...,Xr are brought to such
a form that they are functions of a^, ...,#s only, then the
scheme y\ ='/<(&, ••-,2/J, (i= 1, ...,»)
will make the stationary functions of Fl5 ..., F,. functions of
yv...,ys only.
From what we have proved in § 115 as to the form of the
coefficients £fel, ..., £ks in Xx, ...,Xr, we see that these co-
efficients will be the same functions of xv ...,xs that "njaf'iVhs
are of yXi ...,ys ; and therefore, if any function f(x1, ...,xs) is
an invariant of Xx, ...,Xr> f(yx, ...,ys) will be an invariant of
F F
If therefore we reduce each group to its standard form we
may take x1,...,xm, xq+m+1, ...,xn
to be the invariants of Xls ..., Xr, and its stationary functions
to be functions of xx, ...,xs only ; and we may take
V\i "'•>Vmi Vq+m + H •••>2/«
121] GROUPS MAY BE SIMILAR 151
to be the invariants of Yt, ..., Yr, and its stationary functions
to be the same functions of yla ...,ys, that the stationary func-
tions of the first group are of x1, ...,x8.
§ 121. Let us now say that the g-fold in x space
(1) X1 =i a1} ...,Xm = am, %m + q + i = Um + q+i> •••5#« = an
corresponds to the g-fold in y space
(2) 2/l = alJ •••' Vm = am> Vm+q+1 = Jm+q+l> •••> Vn = /w>
where /m+g+1, ...,fn are any (n — m—q) fixed functions of their
arguments a1,...,am, am+q+lt ...,«„, such that am+q+1, ..., an
can be expressed in terms of a1} ..., am and ym+q+x, •••>2/»*
We have now established such a correspondence between
'the two g-way loci, that when one is known the other is
known.
Under the operations of the group X1,...,Xr all of these
g-folds in x space are invariant ; and if on one of these we
select any point P by an operation of the group Xlt ...,Xr
P can be transformed to any other point on the same g-fold.
Similarly the g-folds in y space are each separately invariant
under the operations of the group Yls ..., Yr ; and by a suitable
operation of this group any point on one of these ^-folds can
be transformed to any other point on the same g-fold.
We now wish to establish a correspondence between the
points in two corresponding g-folds, one in the x space and
one in the y space.
We take as the 'initial' point on (1) the point P whose
coordinates xm+1, ...,xm+q are all zero; and we take as the
' initial' point on (2), which is to correspond to P, the point
Q whose coordinates are
Um+X = Oj •••52/s = 0) 2/s+l ~/s + l> •••' Vm+q = Jm+q
(we proved in § 117 that m + q<£s), where fs+l, ...,fm+q are
any fixed functions of their arguments,
alJ-"'am' am+q+l> •••' an'
We have now established a correspondence between the
' initial ' points on any two corresponding (/-folds ; we get the
correspondence between the two spaces by the convention
that the points obtained by operating on the coordinates of P
with eeLxl+...+er2rr
152 A CORRESPONDENCE [121
shall respectively correspond to the points obtained by opera-
ting on the coordinates of Q with
eelYl+... + erYr^
There are ' initial ' points P lying on each of the g-folds in
x space; to take P, a point on any one particular 5- fold, would
merely establish a correspondence between the points of that
q-fold and the corresponding (/-fold in y space ; by taking
initial points on each 5-fold we have the complete corre-
spondence between the two spaces.
It must now be proved that we have established a point-to-
point correspondence between the two spaces ; i. e. the doubt
must be removed as to whether the operators
ffi\ X\ + • ■ ■ + er xr arid gei -^1 + • • • + <r xr
applied to the point P might give the same point in x space,
whereas the operators
(pLYl+...+arT, ^a eei:Fi+...+«rrrj
applied to the point Q might give two different points in y
space.
If e^xv + ... + erxr and ee1x1 + ... + 6rxrj
applied to P give the same point, then the operator
e—elX1 — ...—erXr ee1Xl+ ... + (rXr
will not alter the coordinates of P at all ; that is, this operator
will belong to the group of P.
By the second fundamental theorem (§ 50)
e~eiXl-...-erXr gf1J1 + ...+(9.Zr _ €klXl+ ...+\rXr
where A2, ..., Xr are constants, which are functions of
°\ 5 • • • ? °r ' ^1 ' • • ' > *r '
and the structure constants of the group Xx,...,Xr; and
therefore, as these structure constants are the same for the
group Ylt...,Yrl
e-e1F,-...-crrr e<1rl + ... + «rFr _ e\Yl + ...+XyYr^
The doubt which we have suggested as to the unique corre-
spondence will be removed when we prove that if
Aj Xx + ... +Xr Xr
122] BETWEEN TWO SPACES 153
is an operator of the group of the point P with respect to
X1} ..., Xr, then
A.j Y j + . . . + Kr Y r
will be an operator of the group of the point Q with respect
Since A2 Xx + . . . + Ar Xr is an operator of the group of P,
we have by § 111,
j = r—q
*& + 2 Xq+j <Pq+j,k(X°l> •"> O = °> (* = J5 — > ?)>
where x%, ...,a;° are the coordinates of P.
Now by hypothesis the functions Qq+j^, ... only involve
the coordinates x\, ...,xs; and if the coordinates of Q are
Vi, ...,<, we have y\ = x», ...,y°s = aPs, so that
h+^2\+j<t>q+j,k(yv->y°n), (*= if .».«);
and therefore AjF,-*- ... + ArFr is an operator of the group of
Q with respect to F15 ..., Fr.
§122. We have therefore established a point-to-point
correspondence between the two spaces ; it may be noticed
that, having proved that the coefficients of 5^-' •"'5; — *n
Xls ...,Xr are the same functions of xlt ...,xs that the corre-
sponding coefficients of ^ — » • • • j -^ — in Fz , . . . , Fr are of 2/15 ...,ys,
it will now follow that, if yla ...,2/„ is the point in 2/ space
which corresponds to xx, ...,xnin x space, we must have
Let $ denote the transformation scheme which transforms
any point x1} ..., xn to the corresponding point yx, .-.,yn in the
other space, then 8f{x1,...,xn) will be equal to f(y±, ..., yn)
where / is any function of its arguments.
We take P to be the 'initial' point on any q-fold in x
space ; by varying the coordinates of this g-fold, and the
parameters els ...,er in the operator
this operator applied to the coordinates of an initial point P
will transform it to any point in space x.
154 OPERATORS PERMUTABLE WITH [122
We may say then that
ee1X1 + ...+e,.Xrp
will be a general expression for any point in the x space.
The point in the y space which corresponds to this will be
eeiYl+... + erYrQi
and therefore
geelXl + ...+erXrp _ ee1rI + ...+crrPQj
Or, e-e1Y1-...-e,.Yr geelXl + ...+erXr p _ Q
We now take another independent set of parameters
ep ..., er, then
Since eei^i + -- +e»^r p is any point in the aj space, we must
then have the identity
e€lr1 + ... + 6rrr e_e1r1-...-e,.rr>§
and by the second fundamental theorem we therefore have
where A15 ..., Ar are constants which are arbitrary, for they are
functions of the structure constants, and the arbitrary con-
stants e15 ..., er and e15 ..., er.
Since we have now proved that
gXt rx + ... +Xr Fr _ ^gfgXi^i + ... + xr-^r £-*,
we see that the groups are similar ; and that they are trans-
formed into one another by the transformation scheme S ; and
that the operators Xx, ...,Xr are respectively transformed to
Y Y
§ 123. A very important theorem may almost immediately
be deduced from the proof of the foregoing theorem on the
similarity of groups ; to obtain it, however, it is necessary to
consider closely the form of the transformation scheme S,
which has converted the points of the x space into the points
of the y space.
This theorem is the answer to the question which now
123] THE OPERATORS OF A GROUP 155
arises, viz. what are the transformations which will transform
each of the operators of a given group into itself ?
We might put this question thus, what are the transforma-
tions which will transform
(!) Xh = £fci^r +••• + &» a^T' (&=1, ..., r)
into
(2) Yk = rlkity-+- + Vkn^> (k=l,...,r),
where Xv ..., Xr are the operators of a group, and r]hi is the
same function of yx, ...,yn that £ki is of xx, ..., xn1
Suppose that Xx,...,Xg is in standard form; we take to
correspond to the g-fold in x space given by
(3) Xx = Clx, ..., Xm = dm, #m + g+i = am+q + l> '••' xn = an>
the g-fold in y space given by
(4) Vi = ai> •••) Vm = am> Vm+q+l = ^m+g+l + ^m+g+l' ••■>
where £s+1, ..., £n are small constants which will not vary
from ^-fold to g'-ibld in space y.
To the • initial ' point P on (3) we take as correspondent on
(4) a point Q, whose coordinates are
Vrn+l = 0' •••' Vs — 0j 2/s+i — ^s+l' •••' Vm+q = ^ra+q'
If we now establish the correspondence between the two
spaces we notice that the coordinates of Q differ infinitesimally
from the coordinates of P. Therefore, since Xh is obtained
by replacing the variables y1} ...,yn by xli ..., xn respectively
in Yk , if P' is the point obtained by operating on P with any
finite operator of the group Xx, ..., Xr, and Q' the corre-
sponding point obtained by operating on Q with the corre-
sponding finite operator of the group Yx, ..., Yr, the coordinates
of P' will also differ infinitesimally from those of Q'.
We now have in this correspondence
2/i = xu •••> Vs = *«'
and also, since xm+q+1, ...,xn are invariants,
Vm+q+l = xm+q+l ' 'm+g+l» •••> 2/» ^n^^n^
and finally
J =?B + 2-S
where Cs+i, s+j, ... are some functions of the variables x13 ..., #n.
156
OPERATORS PERMUTABLE WITH
[123
These equations give (n — s) infinitesimal transformations
transforming (1) into (2); the corresponding linear operators
are Za+1, ..., Zn, where
j = m+q— s ,.
'm + q + i ^x
m + q + i
j = m+q—i
"s + i —2. C +
* 8+J Ix ■ *
(i = 1, ..., n — m — q),
(i = 1, ..., m + 2 — s).
We shall now prove that the determinant
£m+2,s+l> " • *m+q,m+q
does not vanish identically, and therefore conclude that these
operators are unconnected.
When we take xm+1 = 0, ..., xm+q = 0, that is, when we take
«15 ..., xn to be the point P,y1,...,yn will be the coordinates
of the point Q, and therefore ys+l = £g+1, ...,ym+q = *m+g; it
follows that Cs+i, s+ • will then reduce to e^, where, as usual, e^-
is unity if i = j, and zero if i ^ J.
The determinant cannot then vanish identically, since it is
equal to unity when we take xm+1 = 0, ..., xm+q = 0.
Since any infinitesimal transformation which transforms (1)
into (2) must transform yx into x±, ..., ys into xs) we see that
there cannot be more than (n — s) unconnected infinitesimal
transformations which have the required property.
§ 124. We have now found (n — s) unconnected operators
Zg+1, ...,Zn which have the property of leaving each of the
operators Xv ..., Xr unaltered in form, and have proved that
there is no operator unconnected with Zs+1, ..., Zn which can
have this property.
Applying the transformation
x^ — Hi + tZfrXj,
we see that
X'j = Xj+t(Zk,Xj),
and therefore the alternant (Zh , X •) must vanish for X'- = X • .
The operators Zg+l, ...,Zn form a complete system of which
(i= 1, ...,n),
0*=l,...,r),
125] THE OPERATORS OF A GROUP 157
the invariants are the stationary functions of Xx, ..., Xr;
suppose now that
£• = n — s
\"s+ii ™a+j) = -2* Ps + i, 8+j, s + k"s + k>
where ps+l s+j, s+k, ... are functions of x1}...,xn.
Since Xm is permutable with Zs+i and with Zs + j, it follows
from Jacobi's identity that it is permutable with the alternant
(Zs+i, Zs+j) ; we therefore have
k = n — *
2* (Xm ps+i, s+j,s + k) Zs+h = ° >
and therefore, since Zs+l, . .., Zn are unconnected, each of the
functions ps+i,s+j,s+k> ••• is an invariant of the group
Y Y
Suppose now that Xx, ..., Xr is non-stationary; we see
that there are no operators leaving the forms of the operators
Xl,...,Xr unaltered; there are therefore no operators per-
mutable with each of these operators.
If on the other hand Xlt ..., Xr is stationary there are
(n — s) such operators, viz. Zs+1, ..., Zn; these will form a
complete system
jfc = n — s
\^s + ii "s+j) = *-* Ps + i, s+j, 8 + k"s+k>
of which the structure functions ps+ij g+/,s+&, ••• are invariants
of X1? ..., Xr; if then Xx, ..., Xr is a transitive group, these
structure functions must be mere constants, and Zs+1, ..., Zn
will generate a group which will be finite and continuous,
and have all of its operators unconnected.
§ 125. Suppose now that the group Xlf ..., Xr is simply
transitive ; it is then stationary, for the stationary functions
vanish identically ; and in it 8 = 0 and r = n ; it will now be
proved that the simply transitive group Z1,...,Zn has the
same structure as the group Xls ..., Xn.
We may take as the n independent operators of Xlt ..., Xn
■s i*. = v = n ~
(1) Xk=5—-+^lhkftvxli5—+...s (k=l,...,n),
o Xf- o Xv
where the terms not written down are of the second or
higher order in powers and products of xx, ..., xn.
158 RECIPROCAL GROUPS [125
We may similarly choose as the operators of ZXi ..., Zn
H=v=n -
(2) 2'& = -^ + 2^v^^+.»5 (*= If ...»»),
where A*^, ..., 4>„, ... are sets of constants.
since (xj;^) = o, (;:;;;;;;:).
we must have
2 (for + ^i*F) jf— + ... = 0,
where the terms omitted are of higher degree than those
written down.
This identity gives
(3) lkiv + kkv = 0, (!=J""'%; *=1,. ..,»).
v& = 1, ...,w; /
We also see that
(Ari5 X*) = 2 (hiv — ha.v) — + ... ;
and therefore the structure constants of X1} ...,Xn are given
J Cikv = tlkiv — 'likv
Similarly the structure constants of the group Zls ..., Zn are
° I* Cikn=- hkv — hiv'i
and therefore by (3) we see that the two groups X15...,Xn
and Z1,...,Zn have the same structure constants when we
take the independent operators in the respective forms (1)
and (2).
The two groups Xls ..., Xn and Zlt ..., Zn are said to be
reciprocal to one another.
CHAPTER XI
ISOMORPHISM
§ 126. We have proved in § 58 that the structure constants
of a group are the same as those of its parameter group ;
we shall now give a second and more direct proof of this
theorem.
If afi = (pi'zi + -+arjrrxi, (i=l,...,n)
are the canonical equations of a group, then we know that
/j\ &alX1 + ... +ar Xr eb1X1 + ... + brXr __ gq -Z\ + ... + crXr
where clf ..., cr are functions of al5...,ar, b1,...,br, and the
structure functions of the group.
Let ck = Fk(a1, ...,ar, bx, ...,br), (k as 1, ...,r),
then 2/7, = Fk (2/i> •••> Vr> <h> •••> ftr)> (k = J> — > r)
are the equations of the first parameter group in canonical
form ; and the equations of the second parameter group are
yyk = Fk(a1,...,an y19 ...,&.), (k = 1, ..., r).
The forms of the functions F1,...,Fr are fixed by the
identity (1), and can be determined in powers and products
of «!, ..., ar, 619 ..., br when we merely know the structure
constants of X15 ..., Xr\ the method of obtaining these func-
tions is partly explained in Chapter IV, and more completely
in a paper in the Proceedings of the London Mathematical
Society ,Vol. XXIX, 1897,pp. 14-32. As, however, we now only
require the expansion up to and including powers of the
second degree, we shall obtain this expansion from first
principles.
Neglecting, then, all powers above the second, we have
e»xebY=(l + aX+jX2)(l + bY+^Y*),
= l+aX + bY+jX* + abXY+^Y*;
160 THE PARAMETER GROUP [126
and therefore, since
(aX + bY)2 = a2X2 + ab(XY+YX) + b*Y*,
eaXehY=l+aX + bY+\{aX + hYf + hab{XY-YX).
This is true whatever the linear operators X and Y may be ;
and therefore the identity (1) gives
l+c1X1+...+crXr + \(c1X1+... + crXr)2
— 1 +(a1 + b1) X1 + ... + (ar + br)Xr
+ 1 (K + b1)X1+...+ (ar + br) Xrf
+ i2 (ai bj ~ aj bi) (Xi> Xj)-
To the first approximation we therefore have
In order to obtain the next approximation we substitute in
the terms of the second degree ak + bk for ck, and, by aid of
k = r
the identity (Xh Xj) = 2C#& Xki
we thus obtain
i=j = r
ch = afc + 6ft + *2 (ai bj ~ aj hi) Cijk + "••
From this we see that the first parameter group is
i=j = r
y'k = Vk - ak + *2 (Vi aj-Vj ai) ciji* + • • • •
The identical transformation is obtained by taking
ax = 0, ...,aB = 0;
i = r
and then ^ = tkj + 1 2 cijh Vi>
where €k- has its usual meaning.
§ 127. The infinitesimal operators of the first parameter
group in canonical form are therefore
where the terms not written down are of higher degree in
yt, ...,yr than those written down.
127]
AND ITS STRUCTURE CONSTANTS
161
Since Fx, ..., Yr are the operators of a group we can, with-
out any further calculation, find the structure constants of
this group ; for suppose that
k = r
(Yi^j)=^dij1iYli,
we verify at once that cijk = d^.
If we were to obtain the complete expansions for Yx, ..,, Yr
we could verify the group property ; and thus prove directly
the third fundamental theorem, viz. that a simply transitive
group can always be found to correspond to any assigned set
of structure constants. All that we have attempted to prove,
however, is that, Fl5 ..., Yr being known to generate a group,
that group has the structure of the group Xlt ...,Xr.
Similarly we may see that the operators of the second para-
meter group in canonical form are
^• = w-*2<^&S^ +
tyj
U = l,...,r).
We know that these groups are simply transitive ; and any
operation of either is permutable with any operation of the
other : they are therefore reciprocal groups, and we may easily
verify that the structure constants of
Y1,...,Yr and —Z1, ..., — Zr
are the same.
When we were given the finite equations of a group
*» — Ji\xV •••' xiv ai> •••' ar)> (* == 1j'"j%)>
we found (§ 40) definite operators corresponding to the para-
meters a19 ..., ar, and we denoted these by
Any operator, however, dependent on these is equally an
operator of the group ; and when we are given any r inde-
pendent operators X1,...,Xr we can pass to another set
Yv ..., Yr, where
Yh = hi *! + ... + hr xr> (k= 1,..., r),
and take these as the fundamental operators of the group,
provided that the determinant
h
115
h
!/•
does not vanish.
CAMPBELL
hrl,
M
h
TT
162 ISOMORPHISM, SIMPLE [127
When therefore we speak of the canonical form of a group,
we mean the canonical form corresponding to some one given
set of operators X1,...,Xr. If we pass to a new set of
operators we change the canonical form of the group ; and
therefore change the corresponding canonical forms of the
parameter groups, by thus introducing a different set of
structure constants.
§ 128. If we have two groups
(1) x\ = e«i^i + ...+«,-^iC.j (2) 2/'. = ea1Y1 + ...+arY,.yii
and if we denote by Sai, ..., ar that operation of the first
which has the parameters a1,...,ar, and by Tax,...,ar the
operation of the second with the same parameters, we say
that Sai,...,ar and Tai, ...,ar correspond.
It does not follow that, if Sai, ...,ar and Sblf ...,br are two
operations of the first group, and Tai, ...,ar> Ti1,...Jbr the
corresponding operations of the second, the operation^, ..., cr
will correspond to Tyx, ...,yr, where
^Cd •••> Cr == ^#1? •••» dr £>bn •••? Or
and -*yij •••» yr = -^^i> •••>cir -* ^i» •••» Or*
This is only true if y1 = cx , . . . , yr = cr ; that is, if the two
groups have the same parameter group.
Two groups are therefore then, and only then, simply iso-
morphic when they have the same parameter group.
Two groups, of which the fundamental set of operators of
the first is X15 ..., Xr, and of the second is Y1} ..., Yr may not
have, with respect to these operators, the same parameter
group ; and yet they may be thrown into such a form that
they will have the same parameter group.
If we can find r independent operators, dependent on
Fj, ..., Yr, and such that they have the same structure con-
stants as Xv ... ,Xr, then, with respect to these new operators,
the group Yv...,Yr will have the same parameter group as
Y Y
Two groups of the same order
x'{ — eaixi + — +arXrXi and y'i = eaiTi + '- + arTryi}
are therefore then, and only then, simply isomorphic when
the two sets of operators Xlf ...,Xr and Yx, ..., Yr have the
same structure constants.
§ 129. Having explained what is meant when we say that
two groups are simply isomorphic, we shall now consider the
129] AND MULTIPLE 163
analogous relation as to isomorphism of two groups whose
orders are not the same.
Let (1) x'i = eai*i + - + arXrXi
be a group of order r, and
(2) 2/. = ^ + ...+^.
a group of order s, where s < r.
These groups may or may not be groups in the same
number of variables ; we establish a correspondence between
the operations of the groups thus ; we take
ak — ^&iai+ ••• +hkrar> («= 1, ..., s),
where hk;, ... are a set of constants such that not all s-rowed
determinants vanish in the matrix
n j * * *
]/•
"'si* ' ' ' sr
and we then say that the operation Tai, ..., ag in the second
corresponds to the operation Sai, -..,ar ^ the first.
The first group is now said to be multiply isomorphic with
the second, if the constants hkj, ... can be so chosen that,
whatever the values of the parameters aXi ..., ar, blf ..., br,
the operation Tai, ..., as Tplt ..., p8 corresponds to the opera-
tion Sai,-.',ar Sb1,...,br, where J3k is the same function of
bx, ..., br that ak is of «ls ...,ar.
We know that a1 = 0, ..., ag = 0 are the parameters of the
identical transformation in (2); suppose that «15 ..., ar, b1,...,br
are two sets of values of parameters satisfying the equations
(3) 0 = hlclyl+...+h1iryr, (k = 1, ..., s),
Since the identical transformation in (2) corresponds to
Sai,..., ar and also to 8bt ..., hr, if the groups are isomorphic
the identical transformation will also correspond to SCl, ..., cr>
where Sclt ..., er = Sai, ..., or Bii> •••> *«■> ^ therefore
0 = hklc1+ ...+hkrcr, (k = 1, ..., s).
It follows that all the operations Sai, ..., ar where ax, ..., ar
are parameters satisfying the equation (3) form a sub-group
of (1).
We shall next prove that this sub-group is self-conjugate.
m a
164 GROUPS MULTIPLY [129
Since (1) is in canonical form, the inverse operation to
Sai, •••) ar is S-a\, ••-> -ar ; that is,
w ai> •••j ar = &-ai> •••> — aT'
Let Shi, •••> 5r be any operation of (1), and T7^, ..., pr the
corresponding operation of (2) ; then to S^fa, ..., br there will
correspond T'1^, ..., pr in (2). Therefore if ax,...,ar are
the parameters of the sub-group the corresponding operation
to /S$j, ..., br San "-iOr ^~Jh> •••> #r must be the identical one ;
and therefore S/Jl,...,br Sai,...,ar S'1/^, ..., lr is itself an
operation of this sub-group, and therefore the sub-group is
a self-conjugate one.
§ 130. We may simplify the further discussion of the
isomorphism of the two groups by taking Xs+1, ..., Xr to be
the operators of this self-conjugate sub-group. The equations
(3) of § 129 must then be satisfied by yx=^ 0, ..., yg = 0, and
Vs+i' •••> Vr may be taken arbitrarily : it follows that we must
now have h} • = 0 if j > s.
The equations which establish the correspondence between
the operators of the two groups are now
ak = hiai+---+hsas> (* = ^—j s) 5
and it is easily seen that by taking a new set of operators,
dependent on the first set X1, ..., Xs, we may still further
simplify these equations, and throw them into the form
ak — ak> (^ = !j ■■■!*)■
Since the first group is multiply isomorphic with the second,
ealY1 + ...+asYSeblYl + ...+bsYa an(j enlXx + ... + a,.Xr e\ Xt + ... + brXr
must correspond ; and therefore, by considering the form of
the functions e1,...,cr given in §126, we can see that the
structure constants of Yx, ..., Ys are given by
k = s
(Y*> Yj) = 2 cijh Yk> (a . . i ' ) 5
that is, the structure constants of F]5 ..., Ys are the same as
those of X1, ..., Xs if we only regard the coefficients of
Xly ..., Xs and not those of Xs+1, ..., Xr in the alternants
<^j>. 6z\"-\>
Unless , then, a group has a self-conjugate sub-group it cannot
131] ISOMORPHIC 165
be made multiply isomorphic with any group of lower order,
except the group of zero order which consists merely of the
identical transformation. A group which contains no self-
conjugate group other than the group itself and the identical
transformation is called a simple group, and therefore a simple
group cannot be multiply isomorphic except with the identical
transformation.
§ 131. When we are given the structure constants of a group,
we can find the structure constants of every group with which
the first is multiply isomorphic.
We shall see later on that, given the structure constants
of a group, all the groups of such structure may be found ;
we now anticipate this result, and assume that, knowing the
structure constants, we know the operators X1,...,Xr of
the group. There is no real need of the knowledge of these
operators in the proof of the above theorem on isomorphism ;
it is, however, more simply expressed by aid of these operators.
Assuming, then, that we know the operators Xls ..., Xr we
find a self- conjugate sub-group, and take its operators to be
We now have
(xi> xj) = 2 cijk xic> ( o _ i ' ' «,) '
J — i, . .., o
and therefore
k = s t — r—s
(^m> {Xi> Xjj) = i Cijk (Xml Xk) + 2* c i,j,s+t (^m> Xs+t).
Since Xs+1, ...,Xr is a self-conjugate sub-group, if we now
apply Jacobi's identity to any three operators of the set
XJt ..., Xs we can verify that
are a set of structure constants of order s.
If Fj, ..., Ys is a group of order s with these structure
constants, then X1,...,Xr will be multiply isomorphic with
Fl5 ..., Ys; and in this way we obtain all groups with which
X15 ..., Xr can be multiply isomorphic.
We may exhibit in a tabular form the relation of the two
groups somewhat as in the Theory of Discontinuous Groups
(Burnside, Theory of Groups, § 29).
If ea1x1 + ... + arxr jg any finite operator of the group, of
which Xs+1, ..., Xr generate a self-conjugate sub-group, we
166 GROUPS MULTIPLY [131
form a row containing this operator by allowing alt ..., ag to
vary, and keeping ag+l,..., ar fixed ; and we form the column
containing this operator by allowing as+1, ..., ar to vary, and
keeping ax, ..., as fixed.
If we take any row, and write in it ag+1 = 0, ..., ar = 0,
and replace Xx by Fx, ..., Xg by Yg) we have the finite opera-
tors of the second group ; and to any two operators of the
first group found in the same column only one operator in
the second group will correspond.
§ 132. Suppose next that we are given a group Xx, ..., Xr
of order r such that
/.- = ?•
and that we are also given r other operators Ylt ..., Tr such
that (^JFi)=2^ftFfc;
and suppose further that only s of these operators are inde-
pendent, viz. Yls ..., Fg, and that
¥s+j = "'s+y.i *i + ••■ +'^+y, s -*«> w — *s •••»*' — s).
If now instead of X15 ..., Xr we take any other set of inde-
pendent operators Xx, ,.., Xr, dependent on the first and such
that Xk = lhlX1+...+lhrXr, (&= 1, . ..,«");
and instead of Y13 ..., Yr take Ylt ..., Yr where
F& = lui Fx + ... + ^.r Fr ,
then if
(i) (^,^)=2fyi*i.
we must also have
fe=r
(2) (F$, F^) =2%'fc F&.
It should be noticed that though from (1) we can infer (2),
we could not infer (1) from (2).
We can now simplify the relation between the two sets of
operators X and Y by taking as the independent operators
of the group Xlt ...3Xr, where Xx = Xx, ..., Xg — Xs, and
k = s
xs+t = xs+t-^hs+t,kxk> (* = !,...,»•■-«);
133] ISOMORPHIC 167
and we have
If <?#&,... are the structure constants with respect to
Xx, ...,Xr we now see (since Fs+1 = 0) that
and therefore Xg+1, ..., Xr generate a self-conjugate group.
The operators Ylt ..., Fs are now independent, and, since we
have (Ti} Xj) =^cij1eTh, CZ-.'""8*)*
and ffi 1J) «2 Vi ** (Jl I' ■"'!)■
we see that Xx, ..., Xr is multiply isomorphic with Y1} ..., Fs,
the independent operators of the set Y1,...,Yr; and that
X8+1, ...,Xr, the self-conjugate sub-group, corresponds to the
identical transformation in the group of order s whose opera-
tors are Ylt ..., Yg.
§ 133. We had an example of isomorphic groups when we
proved in § 104 that the contracted operators, with respect to
any equation system which admitted the group X1,...,Xri
had the same structure constants as the operators Xlf ..., Xr.
If the number of independent contracted operators is r, the
isomorphism is simple ; but if the number is less than r then
Xv ..., Xr is multiply isomorphic with the group of its con-
tracted operators.
Example. Prove that the group Xv...,Xr is simply or
multiply isomorphic with E1,...,Er where
j = » = r
Eh =^cjhsej^ * (k=l, ...,r),
according as Xli...iXr does not, or does contain Abelian
operators.
Example. Prove that if two transitive groups are simply
isomorphic in such a way, that the sub-group of one, which
leaves a point of general position at rest, corresponds to the
sub-group in the other, which leaves the corresponding point
168 SIMILAR GROUPS [133
of general position at rest, then the two groups, if in the
same number of variables, are similar.
The equations which define the groups of x®, ..., x°n and
3& •••>#« are respectively (§ 111)
j = r—n
Ck + lLen+jtn+jM^ — »*£) = °> (k = l'* »•*»)■
and
j = r—n
and therefore, since e4- = e^, we must have
We have proved that
■X- i Vn+j, k = **i, n+j, k '
and therefore, if X\ denotes the operator obtained from Xk by
substituting for xx, ...,xn the respective quantities x\, ...,£°,
and 4>°n+j,k? ui,n+j,k denote respectively the functions <£„+,-, &,
^i,n+j,k with x\, ...,a?°, substituted therein for xlt ...,xn, we
have ^°i€+j,k = ^ln+j,k'
Now since the two groups are simply isomorphic and
ft+j,k = V#+/,ft> we must have
and therefore, since
we must have
<\>n+j,k{Xv~->Xn) = ^n+j,k(yv~>yn)> ( & _ 1 ' J*
The groups therefore satisfy the sufficient and necessary con-
ditions for similarity.
CHAPTER XII
ON THE CONSTRUCTION OF GROUPS WHOSE
STRUCTURE CONSTANTS AND STATIONARY
FUNCTIONS ARE KNOWN
§ 134. In Chapter X we proved that two groups are similar
when they have the same structure constants and stationary
functions. In this chapter we shall show how when these
constants and functions are known the group may be con-
structed.
WTe take the case of transitive groups first; let Xl5 ..., Xn
be unconnected and
0) Xn+j=^2<Pn+j,kXk> (j =l,...,r-7l);
suppose that s of the stationary functions are unconnected,
and that these are functions of a\, ...,xs only.
We saw (§ 115) that
t = n
(xi> Xj) = 2 uijh xk> (j _ 1}' '"] n) s
where U^j.,... are a known set of functions of xx, ..., xg which
we call the structure functions of the complete system
X15 ...,Xn ; and if
xk = £fci^7 +", + ^^T' (k=l,...,n),
we proved that f^, ...,^-s are known functions of xt, ...,xs.
It follows therefore that Xmn^fc, ... are all known functions
The problem which lies before us is therefore to determine
the forms of n unconnected operators in xlt ..., xn, such that
k = n
(Xi, Xj) = 2, ^ijk Xk>
where the structure functions n^-j, ... are known, and also
170 COMPLETE SYSTEMS [134
the functions obtained by operating on these functions with
-^■i -^-jj-
When we have found X1,...,Xn then we shall also know
Xu+lt ..., Xr by (1).
It" 8 = n, that is, if the group is non-stationary, since we
know ^j, ..., ij.s we know Jfj — , Xn at once.
We now assume that $< n so that the group is stationary.
§ 135. If we have any n unconnected operators we know
1 = 71
(§68) that (*»,*,)= 2 P** XkJ
from the identities
(XJiX{) + (Xi,XJ) = Ot
(X,, (Xi; X,)) + (Xt, (X,, Xj)) + (Xk, (Xj, X()) = 0,
we therefore deduce the following relations between the
structure functions p^k, ...
(1) Pijk + PjiJc = °>
f = r
Xj Pvem + XiPkjm + Xk pJim + 2. (Pikt Pjtm + Pkjt Pitm + Pjit Pktm) — °>
where i, j, &, m may have any values from 1 to n.
If the structure functions P;jk,... are mere constants
Xj, ...,Xn is a simply transitive group; and we have shown
in Chapter Y how from a knowledge of these constants the
group itself may be constructed. In the case where X15 ..., X„
formed a group Xmp-ik, ... were all zero; the problem before
us now, when pji}., ... are known structure functions satisfying
the conditions (i), and Xmpjih,... are all known, but not
necessarily zero, is to find the operators X15 ...,Xn.
This problem is therefore a generalization of that considered
in Chapter Y. and we shall show how the results of Chapter Y
enable us to solve it.
Not more than n of the structure functions pjjk, . . . can be
unconnected; if n are unconnected we can express xv ..., xn
in terms of these structure functions ; and therefore, since we
know XTO Pij-j., ... , we know Xm (xj, ...,Xm (xj, and therefore
know the operators Xlf ...,Xn.
We next suppose that only s are unconnected where s < n,
and we may now assume that the variables have been so
chosen that the structure functions only involve x1} ...,xs; if
* . 3
then Xt, = fia r— + ... + £kn
-r ... i t^.jj .
Ci'-l °xn
*
136]
OF GIVEN STRUCTURE
171
we see that £/;1 ■ . . . ? £;,.* are &H known functions of xx. .....'/:,.
and what we have to do is to determine & I+1, ..., £;.re.
If we take
*i = ^7:1 *1+ — + ^Zrn ^n> (* = *» — J r0--
where Xu, ... are known functions of a^, ..., ^ whose deter-
minant
a-w •
. X
: .
lnl>
• • A.
does not vanish ; then Y F will each be connected with
Xv .... Xn and they will form a complete system, so that
;. = .
The structure functions o-,--?:, ... of this complete system
must satisfy equations of condition like (1); they will be
functions of x\. .... xg only, as will also be the functions
}"m a-^j. . ...: and finally if we can construct the one set of
operators we can construct the other set of operators.
We now make use of this principle to throw X1; ..., Xn
into the forms
X- =
d d d
A . - - f.
a
■••+ Q$+j,n'
(/ = !,.. .,*—«).
§ 136. In order to find the operators X1; .... Xn which
satisfy
; = - ,; = i.....?ix
(1) (***/) =3 .-,;■:; *:;. (y=l,...,J
we have to find the set of functions £l7;.
The only equations involving £u. .... £ln. or such of them
as are unknown, are those obtained by equating the coefli-
cients of
cV-"-,
*x.
on each side of the identities
Jc= 1.
I 2 . Z | ) = pj :., Xx + . . . + Pjj:n X.t. [ . _ j" ■ ^ J •
172 A SYSTEM OF [136
We must therefore eliminate £n, ..., £ln from
-^ / lv — ± , . . . , At , . \
Xj £h — %k £ji = ^ Pjkm £mi> \j = 1, ..., n \ ' '"' ' '
and thus reduce the differential equations to be solved to
a set not containing £n, ..., £m.
In the form to which we have reduced X1} ..., Xn
we see that p(u = 0. ..., p#a = 0 ; and thus we see that
£n>---> im cannot appear in any of the identities, obtained
by equating the coefficients of ^ — ' *"' > — ^n 0)> unless &
or y is unity.
The only equations obtainable by differentiation and
elimination from
(2) ^ft £lj — Xi Sty — 2* Pklm €mj> \j =s 1, ..., 71 ' '
which will not involve derivatives of £u, ..., £1B above the
fii'st, are
(3) (X{ , Xh) £ jj — X{ Xx £kj + Xk Xx gy
m;=n m = n
== -**• i — Pklm fern/ -**■ k ^-* Pilm £mj '
Now
xi x\ hj — %k %i £y — xi (xi £kj — xk £ij)
+ i^i, XJ £hj— (Xh) Xj) £y,
and
m = n
( 4 ) Xf £kj — Xh fy = 2p am imj ;
so that by aid of these equations and (1) we see that (3) takes
the form
m = n
2* Pikm (^m £li ~ ^l £mj) ~ 2* £mj (%i Pikm + ^i Pklm + ^k Plim)
(5)
m = n
+ 2* Pikm i^i £mj — Xm £ij) + ^ Pilm (^k ^mj~^m hj) ~ °'
We have, in passing to this form of (3), made use of the
equations , _ 0
Pijm* Pjim — u#
137] DIFFERENTIAL EQUATIONS 173
If we now replace
p = n
-^mflj~^l €mj ®y 2* Pmlp €pj »
p = n
and Xi£mj — Xm€ij Dv ^Pimp€pj>
the equation (5) is such that the coefficient of £„;- is seen to
vanish identically by aid of the equations of condition (1)
of § 135. We therefore conclude that the only equations
of the first degree in the derivatives of £n, ..., £m are the
equations (2) themselves. Any equation of the form (4) we
shall denote symbolically by (i, k). What we have now
proved is, that the only equations of the first degree in the
derivatives of £n, ..., £ln are the equations symbolized by
(1, 2), ..., (1, n).
§ 137. If then we have found any values of £kl, ..., £ftn
(where k may have any value from 2 to n) to satisfy the
equations
« <'•*>• dzl::::>
the equations for £u, ..., £ln, viz. (1, 2), ..., (1, n) will be
consistent *.
By aid of these equations (1, 2), ..., (1, n) we can express
X2 £*,..., Xn£Xj in terms of £UJ ..., £1B and known functions ;
for, assuming that we have solved the equations (1), £kl, ..., £fcn
are known functions if k > 1 .
Now X2, ..., Xn are (n— 1) unconnected operators, in which
I — does not occur; and, since £kl, ..., ^w, where k>\, are
known functions, these operators are known. We can therefore
express - — > • • • > r — in the forms
dx.z oxn
^ — — ^ft2-^2 + ,»'+^fcB^»> (k = 2,...,n),
cxk
where \kj-, ... are known functions of xx, ..., xn.
It follows therefore that, when we have solved the equations
* See a paper by the author on ' Simultaneous Equations' in the Proceedings
of the London Mathematical Society, XXXI, p. 235.
174 CONSTRUCTION [137
(1), we can express the first derivatives of £n, ..., £ln with
respect to x2,..., xn in terms of £n, ..., £lw and known func-
tions ; and in these expressions for the first derivatives
£u,..., flfl will only occur linearly.
In these equations vc1 occurs merely as a parameter ; we
therefore look on xx as a constant, and say that we have
obtained expressions for all the first derivatives of £u , . . . , £m
as linear functions of these unknowns, the coefficients being
known functions of the variables ; that is, the types of equa-
tions to be solved are
j^T — ajhl Wi + ... + ajlm um + ajk,m+i > { fc _ j n ) '
where a-^ are known functions of the variables ; and of
these equations integrals may be obtained in the form of
power series.
The operators Xs+1, ..., Xn form a complete system of order
(n — s), and the structure functions of this system only
involve xlt .,., xs. Since these variables only enter the opera-
tors Xg+1, ..., Xn as parameters we may look on the structure
functions as mere constants; and we can therefore by the
method of Chapter V find these operators Xs+1, ..., Xn.
Xs, Xs+1,..., Xn now form a complete system, and as we
know Xg+1, ..., Xn we may therefore by the method we have
just described find the coefficients
and thus find the operator Xs.
Proceeding thus we may find all the operators Xx, ...,Xn,
and have thus shown how a transitive group can be con-
structed when we know its structure constants and stationary
functions.
§ 138. We can now construct the types of intransitive
groups.
Let Xr, ..., X be the unconnected operators of the group
Xj, ... , Xr which we suppose in standard form.
The stationary functions only involve xx, ...,xg, and, since
VC-^ 5 • • »j &iyyi ) Wl + O + 1 5 • • • J 71 ***® lHV£tricLIl"LSj
OU/m+l oxm+q
Since the invariants only enter Xv ...,Xq in the form of
138] OF GROUPS 175
parameters we may consider Xlt ...,Xq to be the operators
of a complete system in the q variables %m+i,"->%m+q; and.
as we have
(*«.*,) =2 %*x*, (i-\ «).
where IT^ft , . . . and Xm IT^j. , . . . are known functions of the
parameters x1,...,xm and the variables xm+1, ..., xg, we can
construct the operators Xv...,Xq as in the previous theory.
When we have thus found Xx, ...,Xq we can find the other
operators by means of the identities
k = q
Xq+j =^<t>q+j,hXk> U = l,-.;T-q).
CHAPTER XIII
CONJUGATE SUB-GROUPS: THE CONSTRUCTION
OF GROUPS FROM THEIR STRUCTURE
CONSTANTS
§ 139. If X1, ...,Xr are the operators of a group with the
structure constants c^-j., ... we have
cij k + cjik = °>
^ (cikh chjm + ckjh chim + cjih chkm) = °-
If Xq+1, ..., Xr form a sub-group we also have
Ofl-rtif+Mss0' U= l,...,r-g; * ~ *' '-*) ;
and if this sub-group is self-conjugate we have the further
conditions
Since our immediate object is to find the general form of
a sub-group conjugate with a given sub-group, it will be
convenient to take a set of operators Y1 , . . . , Yr dependent on
Xx, ..., Xr and defined by
fornrn
|1«
Si
(1) Yk =Xhl (k=l,...}q),
/* = <?
(2) Yq+t = Xi+t-^hq+t^X^ (t = l,...,r-q).
The identities (2) can be written
and therefore, whatever values the constants hq+t!lx,... may
have, Fp..., Fr are independent operators.
It
a!
140] NEW STRUCTURE CONSTANTS 177
If we suppose that hitL = 0 when i > q, or when // > q, the
formulae (1) and (2) may be replaced by
p = q
Yi = Xi-^h^X^, (i=l,...,r).
§ 140. We now introduce a set of functions of these constants
hq+t,p, ... denned by
t = r—q p. = q p. = q
( 1 ) Hijk = cijk + 2 ci,j, 2+t \+t, k + 2 cm« hi* + 2 CJ^ ^»M
ju. = i/ = 2 ji = 5, t = r — 5 n = q,t — r—q
+ ^ C^pt «j^ «•_,-„+ ^ C^,hq + t'ljii hq + t,k+ 2* CJ,H,V + t «*n"'q+t,k
fi. = v = q,t = r—q
Since
(F„ F,) = (X,, X,) + 2 K (Xk> XJ + 2 **n (^> ^)
fi. = v = q
+ 2 ^ ^^ (^s*j ")»
* = r A = r j = 9
and (X„, X,) = 2 W x* = 2 «V»* (Y* + 2 ^ F,),
we see that the structure constants of Y1, ..., Fr are the set
-"{/& > • • • «
It therefore follows that
Hijk + Ejik= °i
(2) * = r
Since X15 ..., Xr are derived from F1} ..., Fr by the law
Xt= F,+ 2V^, (i= l,...,r),
and .H^k, ... are the structure constants of F1} ..., Fr, we must
have
t = r—q H = <i M = 2
(3) c;jk = Hijk — 2 Hhj,q+t hq+t,k—^ H^k hj^ - 2 #>* h^
t=r—q,n = q t = r—q,ii. = q
+ 2 Hn,i,q+thjiJi hq + t,k + 2 *Vil*i2+* "^ ^2 + *,*
jt = v = 2 ft = v = g, i = r— g
+ 2 "Sj"'* ^»> ^'v ~~ 2 ***»■ ".?+' ^*M %» hq + t, k '
CAMPBELL N
178 THE OPERATORS OF [140
Let
t = r-q /u = 9 fx = q,t=r— q
(4) ITyx- = CM + 2 ci,M+t K+t,k + 2 cn» hJlx + ^ c^q+t hJlx kq+t)k,
then we see that
(5) Hijh= riafc— ^hj^Hnjk,
and therefore, since 7i(> =0 if i > <?, £?#* = FTy* if i > g, and
(5) can be replaced by
(6) U;Jk = H;jk + 2 ^*f* ^Wi •
It will be noticed that though i^-fe + 1/^ = 0, IL^ + FT^
is not zero if either i or j exceeds q.
lik>q, H^ takes the simpler form
§ 141. It is now necessary to prove the formula
( = r t = r
(1) ^ (n^.g+^illytt— HV)q+jlt n^a-) =^CVH.t Htlq+j,k-
From (2) of the last article we see that
— \Hp,q+J,t Hvtk —■HVt(l+jttH.IJ.tk) = 2* "v^t -Ht,q+j,k-
If we apply the formula (6) of § 140, we see that
^ (nM, q+j, t n„o,. — nV}q+j} t + u^k)
t = r p = q p = q
=■ 2* (-H-H.i+J, t + 2* *W Hp,q+j, t) {Hvtk + 2* 'l»P Hptk)
t = r P = <1 p = q
— 2* (■">'» q+J, t + — "'"p Hp,<i+J, *) {H^tk + ^ h^p Hptk)-
Multiplying this out and applying (2) of § 140, we see that
it is equal to
t = r t = r,p = q t —r,p = q
2* ■" VH t Ht, q +j, k+ 2L"-HP H-vpt Ht, q +j, k+ ^Kp Hpii | Ht, q +j, k
t=r, p=zp'=q
"J" ^4 "pp hvp' -tlp'pt -tlt,q+j,k'
142] AN ISOMORPHIC GROUP 179
We now replace Ht„+jtli in this expression by
i = q
and we see that, if i > q, the coefficient of nijg+J-)& is the
expression for cVfli in terms of hq+t k, ... and the functions
B#&,... given in (3) of §140.
If i > q this coefficient is
p = q p = q p>=p = q
and if we notice that hu is zero when i > q, we shall see
that this is also equal to cVILi. We have thus verified the
formula (1).
§142. We now look on hq+tki ... as a set of variable
parameters ; since every term which occurs in n^fe either
begins with j or ends with k, we see that, if j > q and fc > q,
3T^ = -niMi and M-J-=ni,j,q+f
We now introduce a set of r linear operators ni5 ..., n,.
defined by
k=q,j=r-q
nM = 2nM,
q+j,k
*kq+J,h
when we have
t=q t= r—q
^tt^lhi+j,t — "~ ^i^V,q+j,t^li.tk+ 2, ^v,q + t,k^^.,q+j,q+t,
t=q t=r—q
11^ Hw,q+j,t = — 2, ^n,q+j,t H„a- + 2* ^(i,q + t,k ^v,q+j,q + i i
and therefore
t = r t=r
n„ ^y.,q+j,h— n^ Ei^g+^t = — _2L n„,q+j,tTintk+ 2* nM,«+i.< **„«*
by the identity (1) of § 141.
It therefore follows that
k = r (% = 1 ... r\
(nlln;.)=2^fcnis ^= i|"^r)'
N 2
180 A SYSTEM OF EQUATIONS [143
so that FFj, ..., nr generate a group isomorphic with Xlt ..., Xr.
If the operators ni5 ..., Ur are independent the groups are
simply isomorphic, but if they are not all independent
Xx, ...,Xr is multiply isomorphic with nx, ..., nr.
§ 143. Still looking on h +tk, ... as variables, we shall now
prove that the equation system
admits these operators.
If we notice that in Hq+iiq+j>1. every term either ends in k
or begins with q + i or q +j, we shall see that if /x > q
p = q P=Q
■i.k
UltHq+i,q+j,k:=jt/^,q+i,p Hq+j,P,k + ^ -"f*,4+i,i» Hp,1+h'
t = )' — q
+ ^ H^q + t^k J^q+i,q+j,q+t
p = r
+ Ilq+jtq+itp -tlppk)
t = r-q P = r
+ ^ Hp.,q + t,k Hq + i,q+j,q + t— ^ -"</ + »>, P ^P,q+j,^
p = q+l
p=r p=r
— ^ MfL.g+i.y Hp,q+i,k — 2i **q +i,q+j,P ■"**»!>>**
Since the expression in the bracket vanishes identically we
see that n^ Hq+itq+j,k = 0 is an equation connected with the
equation system (1); that is, it is satisfied for all values of the
variables which satisfy (1).
Also since
Hq+i,q+j,k = "q+i,Q+j,t jL> ^q + i, n * V. '1 +J> k '
we conclude that, even when fi > q, the equation
n Hq+i>q+j,k = o
is connected with the equation system (1); so that we have
proved that the system admits the operators [I1, ..., nr.
It will be noticed that the operators ni9 ...,ITr are defined
simply from the structure constants c^, ... of the group, as
are also the equations of the system (1) which admit these
144] ADMITTING THESE OPERATORS 181
operators. The group property of the operators YI1, ...,Ylr
might have been proved without any reference to the group
Xx , ...,Xr, though the labour of the proof was much lightened
by that reference.
§ 144. Suppose now that we have any sub-group of
X15...,Xr whose order is (r — q), and suppose that all its
operators are independent of X1 , . . . , X • we may throw the
operators of this sub-group into the form F +1, ..., Yr, where
Yq+t = Xq+t-^ hq+t^X^, (t = 1, ...,r — q),
and we may then take F2 , . . . , Yr to be a set of r independent
operators of the given group where Yk = Xk if k > q.
Since H^, ••• are the structure constants of Fl5 .,., Fr,
and Yq+1, ..., Yr is a sub-group,
Sq+i>q+j)Jc = 0, (jZl,...lr-ql &=cl> ->?) '
These are therefore the equations in the variable parameters
hq+t, k> ••• which define sub-groups of order (r — q).
YQ+1, ...,Yr will be a self-conjugate sub-group if
H _0 ,i=l,...,r-q; , n
that is, the sub-group will then be invariant under any
operation of the group F15 ..., Yr.
Even when not invariant under all the operations of
Fj, ..., Fr, that is, when not self-conjugate, it may be in-
variant under some of the operators.
It will be invariant under the operations of the sub-group
Y +1, ..., Yr in every case; it will be invariant under the
operations
x'i = eaq-hYq-h + ...+arYr Xi, (i = 1, ...,n)
if, and only if,
„ A = 1, ...,r— q: , , \
The operations which transform a sub-group into itself
must from first principles generate a group, which will con-
tain the given sub-group as a sub-group, and therefore the
operators Y h, ..., Yr must themselves be a sub-group of
F F
182
THE SUB-GROUPS
[145
§ 145. Suppose now that we are given the structure con-
stants Ctfj., ... of a group J],..,,Ir, and we want to find
the structure constants of all possible sub-groups of order
(r — q); we equate to zero the functions H +i >q+j,]c> ••• of the
variables h(l+ti^, ....
If no values of hq+t>IL, ... can be found to satisfy the system
then there is no sub-group of order (r — q), all of whose
operators are independent of X1? ..., X • that is, if there is
a sub-group of order (r — q) at all it must have at least one
of its operators dependent on X1 , . . . , X . In this case we
should take, in order to form the functions H +i q+- Ji, some
other set of (r — q) operators out of the set X1, ..., Xr in place
ofX
2 + 1' •••5
Xr; for there is no sub-group of order (r — q)
which cannot be expressed in some one of these ways.
We see this more clearly if we consider the sub-group
!2+i'
, Yr where
fe=r
aQ + t,~ki •
Yq+t =^ aq+t,h^k> (t — 1, ...,r — q),
being a set of constants.
This sub-group could then only fail to be expressible in
the form
k = q
Yq+t = Xq+t — ^hq+ttJcXk, (t = 1, ..., r—q),
when
a
2+i,2+i»
a
9+1, r
a
r,2+l'
a
r.r
= 0;
and it could only fail to be expressible in some one of the
required forms if all (r — §)-rowed determinants of the matrix
a
2 + 1,1:
. a
2+1, r
ii
r, 1'
a
r.r
vanished ; that is, if the sub-group was of order less than
(r-q).
146]
OF A GIVEN GROUP
183
If on the other hand we find a set of values of h,
q + t,i*.i •••
to
■i = 1, ...,r-q;
satisfy the equations
then Hq+iq+j>q+t, ... will be the structure constants of the
sub-group whose operators are
H = q
We then denote the operators of this sub-group by
Yq+1, ...,Yr and the group itself by Y1, ..., Fr.
The sub-group is of course invariant for the operators
Yq+1, ...,Yr; it will be invariant for
e1Y1+ ...+eqYq
if eiHi,q+i,k + - + eqHq!q+i>h = 0, CkZ\\'''^~qq)-
We therefore, in order to find within what group Y +1, ...,Fr
is invariant, write down the matrix
5.
/r.
2, g+i, & >
om 1 to (r — q), and &
rr
where in any row i takes all values fr<
all values from 1 to q.
Suppose that the values of hq+t^, ... now found are such as
when substituted in this matrix will make all (q — m + 1) -rowed
determinants but not all (q — m)-rowed determinants of the
matrix vanish, then the sub-group Y +1,..., Yr is invariant
for m operators independent of one another and of Y +1, ..., Yr.
The sub-group is therefore invariant within a group of order
r — q + m, and there are only (q — m) independent operators
for which it is not invariant. We say, then, that the sub-
group Y +1, ..., Yr is oi index (q — m).
§ 146. We now wish to find the sub-groups conjugate to
Y +1, ..., Yr, so we must consider what this sub-group is
transformed into when we apply the infinitesimal trans-
formation
(1)
Xi
Xt + tYjXj, (i= 1, ...,n).
184 SUB-GROUPS CONJUGATE [146
Ifj > q the operators Y +1, ..., Yr will be transformed into
operators dependent on Y +l, ..., Yr; we need therefore only
consider the case where/ > q.
We saw in § 76 that, X'k denoting the operator derived from
Xk by replacing x{ by x'{,
/* = ?•
x'k = %h + ^2 cjkv. ^-
Hence we now have, since Yl , . . . , Fr are operators with the
structure constants HJ/cili> ...,
(2) ^ = Fft+f2^FM.
Now Y'+1,...,Y'r are the operators of the sub-group con-
jugate to Yq+1, ..., Yr obtained by applying the transforma-
tion (1); and therefore, since this is a sub-group of order (r — q),
and differs infinitesimally from F +1, ..., Fr, it cannot have
operators dependent on Xv . . . , X . We may therefore take its
operators to be
^q + l~ 2d hq+i^X^, ..., Xr— ^ hrlJL X^,
where K]+j,p- = hq+j,n~t^q+j,n> and \+|/!... are functions
of the variable parameters h +i „_, ... whose forms must now
be determined.
The operators Yfq+1, ..., Y'r are operators of the sub-group
^q+l~ 2 "•g+l,i*-^i*J "-iXr — ^hrlliXr',
that is, of the sub-group
H = q (t. = q
Yq + l + t 2^ + 1,1*. i>, ..., Yr + 2 ^■/> -^fl !
and therefore
*q + i = 2eq+i,q+8\Yq+8 + t2i \+s,M *J*)j (* = ^•••»r_ ?);
where e2+i> g+s, ... are constants.
If we now compare this expression for F' + - with the
expression obtained in (2), and equate the coefficients of
Yq+1, ...,Yr we see that, neglecting small quantities of the
order t, eq+i^ +g is equal to unity or zero according as i is or
is not equal to a ; and therefore we see that
x — rr fs ~ *' •••Jr — q\
r* — xj •■•> y
146] TO A GIVEN SUB-GROUP 185
Since j > q, Hj)q+t)ll = UJtq+s>IJi ; and therefore the constants
K+j j*j ••• which define the sub-group conjugate to Yq+1, ...,Yr
obtained by the infinitesimal transformation
afi = xi + tYjxi, (i =},...,»)
are given by
V -I tu A = l,...,r-q,
"q+i,* — nq+i,n iLlj,q+i,n> V/i = 1, ..., q'
Because the sub-group is invariant for the transformations
xi = xi + tyq+jxit \j _ i m r-.q)'
we see that for such transformations
f I TL
nq + i,H — aq + i,n'
We now want to find the constants defining the sub-group
adjacent to that defined by hq+i!lJL, ... and obtained by the
infinitesimal transformation
x'i = xi+(e1X1+...+erXr)xi) (i = 1, ...,n).
We have
fx. = q j = r-q j=r~q
e1 X1 + . . . + er Xr = 2 («M + 2 eq+j hq+j,n) X* + 2 V; ^2+/ '
and therefore
l- = q
hq+i^n = hq + i,n~ <2*\ek~>r ^eq+j "'q+j,k)^k,q+i,f1'
Now, since
if,+i,,+,> = o, (*=J'-^-?>=i,...,?),
and therefore
"q + i,H- = 'lq+i,^~ 2* ek "fc, 5 + *,^'
= ^ + i,^-(eini + ••• + er nr) ^2+»,l»-
The relation between the groups ET1} . . M IIr and Xx , ...,Xr
can now be expressed in general terms. Let hq+t,v, • •• be a set
of constants. defining a sub-group of Xli...iXr\ then the set
of constants h'+tlLi... which define the sub-group conjugate
to this and obtained by the transformation
x\ = eei*l + -+erjrrxi, (i = 1, ...,n)
186 THE POSSIBLE TYPES
are given by the formulae
,lq + t,H — e "q + t,H> \
[146
t = 1, ...,r— q
\x = 1, ..., q
)
§ 147. In order to find all types of sub-groups of order
(r — q) we therefore proceed as follows.
If no sets of values of hq+ttlt., ... can be obtained to satisfy
the equations
(1) ffsHS+i.* = °. (} 2 1, Zl-ll * = 1 ?)'
no sub-group of order (r — q) exists.
If on the other hand such a set exists, let //"+(/, ... satisfy
the equations (1); we write down the matrix of the operators
Uj, . .., ii r
where in any row all values of j from 1 to {r — q) and all
values of k from 1 to q are to be taken. If when we substitute
for h +f ^ ... in this matrix the respective values h°q+ttfJ., ...
all (sh- l)-rowed determinants of the matrix, but not all s-rowed
determinants, vanish, then the sub-group is of index s; and
the ' point' whose coordinates are h° +f ,*, ... is of order s with
respect to the equation system
0) Hq+i,q+j,k = ° (in the variables hq+t>H., ...)
admitting the operators nx , . . . , I7r .
Since
H=q
(2)
■Hq+i,q+j,k — '■*q+i,q+j,k 2* "q+i,!* *-*H-,q+j,ki
the index s cannot exceed q.
We now find (as explained in § 103) the contracted operators
of ni5 ..., Ur with respect to the equation system which con-
sists of (1) and the equations which define points of orders ;
for both of these equation systems are invariant under the
operations of the group ni, ...,Ur.
Let this combined equation system be
/ *~~ 1 'j* — a
(3) hq+t}IIL = ^+<)fl(/ij,...,7^), ( _ ' ),
where hx, ...,h are some unconnected parameters, in terms
148] OF SUB-GROUPS 187
of which those values of h,l+t^,... can be expressed which
satisfy the combined equations; and let P1,...,Pr be the
contracted operators.
Since X1 , . . . , Xr is isomorphic with Ux , . . . , ITr and F^ , . . . , I7r
is isomorphic with Px, ...,Pr, X1,...,Xr must be isomorphic
with P1,...,Pr; but the isomorphism is simple, only when
Pv ...,Pr are independent operators.
Since the parameters of a sub-group of order (r — q) and
index s are by (3) expressible in terms of Al5 ...,h we call
these parameters the coordinates of the sub-group. From the
definition of a point of order s exactly s of the operators
P1, ...,Pr will be unconnected ; and as these are operators in
the variables h1, ...,h we conclude that p < s, and that there
will be (p — s) invariants, which we may take to be
If then h1}...,hp are the coordinates of a sub-group of
index s and order (r — q), the coordinates of the sub-group
conjugate to this obtained by the transformation
SBj = 6«l-Zi + — +erXrXii (1= l,...,7l)
are given by
^ = eeiA + ...+e,-A-£.5 (i= l,...,p).
Since s of the operators of the group P1 , . . . , Pr are uncon-
nected, we can pass, by the operations of this group, from any
point whose coordinates are h\,...,h°, to any point whose
coordinates are Als . . ., hg , h°8+1, . . ., h°p . Sub-groups of the same
order are therefore divided into classes according to their
indices ; only sub-groups of the same order and index can be
conjugate ; and of sub-groups of the same order and index
only those can be conjugate for which the coordinates
hs+1, ...,h are the same. There are therefore co^-s different
types of sub-groups of order (r — q) and index s ; and corre-
sponding to any one of these types we have cos conjugate sub-
groups.
§ 148. We can apply these results to obtain the stationary
functions of groups whose structure constants are assigned ;
and thus complete the investigation of which Chapters V and
XII formed a part, viz. the determination of all possible types
of groups with assigned structure constants.
Suppose the group Xx, ..., Xr is in standard form so that
xv ...,xm, xm+q+1, ...,xn are the invariants, and the stationary
188 THE CONSTRUCTION OF [14 8
functions only involve xx, ..., xs. If x\, ..., x°n is a point of
general position then the group of the point — that is, the
sub-group of operations leaving the point at rest — is of order
(r — q) ; and the coordinates of this group depend only on
x° ..., x° ; for we have proved in § 112 that the equations
**/j w-t j . . . ) *ASg \Aj
— ryto
define the locus of points whose groups are the same as the
group of x\, ..., x°n.
Now by the operations of the group Xlt ..., Xr, only the
coordinates xm+1, ..., xm+q can vary; and, as there are
(r — s + rn) independent infinitesimal transformations which
leave a^+1, •.., #§ at rest, there will be (r—s + m) infinitesimal
transformations which do not transform the group of
^1 5 • • • 5 '"n
This group is therefore of index (s-m) ; and its coordinates
are expressible in terms of s parameters.
In order, therefore, to find the stationary functions of a
group, when we are merely given the structure constants,
we form the equations defining sub-groups of order (r—q)
and index (s — m); the coordinates, then, of the sub-group
which leaves a point of general position at rest will be ex-
pressible in terms of s parameters.
If the combined equation system is
hq+t,». — ^</+«,(x('iiJ •••> hs), ( J,
fj. — I, . .., ^
then the stationary functions fq+t^ (scj, ..., x°n) will be given by
fq + t,^ \X\, •••) Xn) = 99 + *,^ ("U *••' ""«/"
Since the functions (pq+t,^ (hlt ..., hs), ... cannot be ex-
pressed in terms of a smaller number of arguments, we may
express hx, ..., hs in terms of scj, ..., x%; and by a change of
variables we may take A19 ..., hg to be respectively asj, ..., sc°.
As we can vary x\ x% in any way we like, we see that
we may take the stationary functions to be
t = 1, ..., r — q-,
Qq + t,!*. \X1> •••> Xs)> V„— 1 nJ
/* — y J ••• > I
When we have thus found the stationary functions of the
group X1? ..., Xr we may complete the determination of the
operators by the method explained in Chapter XII ; and if
any group with the assigned structure constants, and the
149] STATIONARY FUNCTIONS 189
assigned numbers s, m and n exists, we can find it by the
method now explained.
Such a group may not exist ; thus if we take r > 3, n = 1,
m = 0 and s = 1, we may, for many assigned sets of structure
constants, construct the functions (j>q+t,^s ••• which express
the coordinates of sub-groups of order (n — 1) in terms of one
parameter; but the operators X1,...,Xr in one variable,
which we should hence deduce, would not be independent ;
for (as we shall prove later), no group whose order exceeds
three can exist in one variable.
§ 149. Example. Find all the sub-groups of order 3 of
the group whose structure is given by
(1) (X2, X3) = X15 (X3, XJ = X2, (X1, X2) = X3,
(X15 X4) = 0, (X2, X4) = 0, (X3, X4) = 0.
We first find the sub-groups which can be expressed in
the form X — A X X — X X X — A X
Al — Al-^-45 -A2 — A2'A4> -^-3 A3^4'
that is, the sub-groups not containing X4 as an operator.
Since (X2 — A2X4, X3 — A3X4) = (X2, X3) = X15
we cannot express this alternant in terms of the operators of
the sub-group unless Aj = 0. Similarly we see that we must
have A2 = 0, and A3 = 0.
There is, therefore, only one sub-group of this form, viz.
the self-conjugate sub-group X15 X2, X3.
Whenever by this method we find only a discrete number
of solutions of the equation system
the sub-groups must be self-conjugate; for if they had con-
jugate sets obtained by the infinitesimal transformation
a£= xi+(e1X1 + ...+erXr)xi, (i = 1, ..., n),
there would be an infinity of sub-groups of the required class.
We next find all sub-groups of order 3 which do not contain
Xx as an operator.
The general method of forming equations for hq+tjfLs... to
define sub-groups of order (r — q) is simplified when q = 1.
If we take X2-£2X1S ..., Xr-^XX
190 EXAMPLES ON THE [149
to be the operators of the sub-group of order (r — 1), then the
equations which h2, ..., hr must satisfy are
SiS = hiHlj-hjHlii (. ~ 1,'"|r)»
where Hy = c^ + cfj-2 &2 + . . . + c^-r A r .
In the example before us
H2>3=1,H,^ = 0, #3,4 = 0, -£rM = 0, H1>2 = h3, H1>3 = -h2;
and the equations defining the sub-group are therefore
h4=0, i+v + V = °-
The sub-group sought has therefore the operators
X2 — icos^Xj, X3 — iGm6X1, X4,
where 0 is a variable parameter and i is the symbol v — 1.
By varying 0 we get an infinity of conjugate sub-groups ;
and as the sub-group is not self- conjugate it must be of
index unity.
By interchanging X1 and X2 we should obtain the system
of conjugate sub-groups
X1—icos<f>X2, X3—isin(f>X2, X4,
these two systems coincide, however, the relation between
the parameters being cos Q cos <f> + 1 = 0.
By interchanging X1 and X3 we get
X2—ico3\}/X3, X1—isin\}/X3, X4,
which also coincides with the first system, the relation between
the parameters being sin 6 sin ^ + 1 = 0.
If we try to find a group in the single variable x which
shall have the structure (1) we must take
%2 = <t>2 (X) Xl> XZ = 03 (X) Xl> Xi = 04 (X) Xl'
We now have the following identities which enable us to
determine the stationary functions
02 (x) = i cos x, (p3 (x) = i sin x, $4 (x) = 0 ;
and we see that the operators cannot be independent, X4 being
identically zero.
Now we know that in general Xx <t>q+t,k = ^i,q+t, k ' anc* *n
this example
ni21 = C121 + C122 K + C123 ^3 + C124 K = h = • Sm X>
U121= C131 + CZ32 K + C133 K + C134 K = ~ K = ~ * C0S X 5
150]
so ths
«
}1!
index
Appl;
a .:■
c
so tin
1
and \
In
bi
Sii
genei
Mth,
The .
if
150] CONSTRUCTION OF GROUPS 191
so that, from either of these two equations, we see that, if
X, = A — - , then £ , = — 1 , and therefore X, = — —, and the
1 ^Zx * 1 lx
group is
X, = — zr- > X9 = i cos x — j X, = ■£ sin a: — > X. = 0.
§ 150. Example. Find the sub-groups of order 2 and
index 2 of the group
(^25 ^3) = X13 (X3,X,) = X2, (X1;X2) = X3,
(XlsX4) = 0, (X2,X4) = 0, (X3,X4) = 0.
We shall only find those which are of the form
-^3 — ^31 -^1 ~~ ^32 ^25 ^4 ~ ^41 -^1 — ^42 ^2 ■
Applying the rule (or otherwise) we find the conditions for
a group are
K 2 0+^,i) -^4, 1^3,2^3,1 = 0.
K, 1 (* + M, 2) ~ K, 2h,lhS,2 = °>
so that 1 + hi j + hi 2 = 0.
We must therefore take (A and 0 being parameters)
h3 ± = i cos 0, A3 2 = a sin 0, h± x = \ sin 0, h4 2 = — A cos 0 ;
and we may directly verify that
(X3— 2 cos flXj — ^sin#X2, X4— Asin^Xj + Acos^Xg)
= — iX (X2—icosdX1 — isliD.6X2).
In order to find the corresponding group in the two variables
x, y we suppose that
%3 = 03, 1^1 + 03,2^2* X* = 04, 1^1 + 04, 2^2-
Since the index is 2 we have s — m= 2 ; and, since in
general s cannot exceed n, in this example, s cannot exceed 2,
so that m = 0 and s = 2 ; that is, the group is non-stationary.
The order of the group of the point x°, y° of general position
is (r—q), and therefore (r — q) — 2; and as r = 4 we must
take q = 2, so that the group is transitive, and Xx and X2
must be unconnected.
We have
03, i fo y) = i cos 6, 4>i} 1 (x, y) = A sin 0,
03 2 (a> 2/) = 2 sm 0> 04 2 (*» 2/) = — ^ cos 0.
192
PARTICULAR CASE OF
[150
We may then by a change of the variables take
03, i (•£> y) = x> and 04, i (*> y) = y>
and therefore
tf>3, 2 (*. y) = i ( l + ^2)*> 04, 2 (a, 2/) = - Wx ( ! + x2) ~ *•
We have
^103,1 = ni31 = -^3,2 7i3 1 = — *0 (! -a^)*!
X104,l = nHl = "A,! ^4,2 = *2^ (1 +Z2H,
^203,1 = n231= i+A23)1 = i+«2,
^2 04, 1 = n241 = *4, 1*8, 1 = «2/'
We then see that
X2=(l+^+^>
Now X4 is identically zero, and therefore there is no group
of order 4 of the given structure, but X15 X2, X3 will with
X4 = 0 form a group of order 3 with the required
structure.
§ 151. When the sub-group whose conjugate sub-groups are
required is of order 1 the equations
Hq+iiq+j,jc = °, CjZ l', '.'.'. !r-?; /c=1'""?)
are satisfied identically, since q = r — 1 .
The variables which define the sub-group are hrl, ...,hr p-1;
and e1X1+ ...+erXr
will be the operator of this sub-group if
H + hrker = °> (k=l,...,r-l).
We therefore take hrli = —, and let
■Cy — 2* CJP* ej ^
ek
151] THE GENERAL THEORY 193
In operating on any function ofhrl,...,hr r-1
so that
d _ d ._ .
e,/ ^— = hH ^— if ? < r and & < r,
.± » tf4<r,
'r * oltrk
rde ~2*rt
Therefore, since
i = * = r_l * = r_l /=r-l
^> = 2 C^> * eJ Nj" + 2 Cr*** «rjf7 +2 Cjn r ^ j~ + Cr^ r 6,. —
^ = J- — 1 j = r-l j=r— 1
= ^ ( ^ CJft* "■# + cy.rk -\-Cprr hrf + ^ Cj^ ,. Jl,j hrk) r-j —
d/lrk
k = r— 1
= 2rV*^7— if /* < r,
we see that in operating on any function of hrl, ..., hr r_1
E^ has the same effect as 17M if /* < r.
Since e^ + . . . + erEr = 0,
— rt 1 "t" ••• ' ^r t—1 t— 1 '
and this operator is equivalent to Ur, since the equations
Eq+i,q+j,k = °
are satisfied for all values of hrl, ..., hrr_1.
Since the coordinates of the sub-group of order one are
the ratios of e±, ...,er, we see that for such sub-groups the
operators rT15 . . . , Ur may be replaced by the known operators
EXi ..., Er , of which we made use in Chapter V.
CAMPBELL
CHAPTER XIV
ON PFAFF'S EQUATION AND THE INTEGRALS
OF PARTIAL DIFFERENTIAL EQUATIONS
§ 152. If xx, ..., xn are the coordinates of a point in n-w&y
space, and
(x'1-x1)p1+... + (x'n-xn)pn = 0
(where x[, ..., x'n are the current coordinates) the equation of
a plane through xlf ...,xn, then we speak of the point together
with the plane as an element of this space. We say that the
coordinates of the element are xl,...,xn, px,...,pn, where
xx, ..., xn are the coordinates of the point of the element, and
px , . . . , pn the coordinates of the plane of the element. In the
coordinates of the plane we are only concerned with the
ratios px :p2 ... :pn ; and therefore in w-way space there are
co2"-1 elements.
Two contiguous elements, x1} ..., xu, pXi ..., pn and
are said to be united if the point of one element lies on the
plane of the other. More exactly expressed, the elements are
united if the point of the second is distant from the plane
of the first by a small quantity of the second order. The
analytical condition for this is
(1) p1dx1 + ...+pndxn= 0;
and therefore, if this equation is satisfied, the point of the
first element is also distant from the plane of the second
element by a small quantity of the second order.
The equation (1) is called Pfaff's equation.
Since the coordinates of an element only involve^, ...,pn
through their ratios, we shall suppose that, when we are
given any equation connecting the coordinates
•^1 j • • • j &n » Pi » • • • > Pn
of an element, it is one which is homogeneous in px, ..., pn.
152] PFAFF'S EQUATION 195
If we have m unconnected equations connecting
£Cj , • . . , Xn , p^ , ..., pn ,
viz.
(2) /«(*!> •••,«»» Pi> —,Pt) = °> (i=l,...,m)
then co2n-m_1 elements of space will satisfy this equation
system ; they will be called the elements of the system.
Two contiguous elements of the system will not however,
in general, be united. The question thus arises, what are the
necessary and sufficient conditions which these equations must
satisfy in order that any two contiguous elements of the
sj^stem may be united ? In other words, what are the con-
ditions that the equations (2) may satisfy PfafF's equation ?
Suppose if possible that, from the equations (2), no equation
of the form f/r ™ \ .. 0
can be deduced ; we must then be able to express m of the
coordinates p1,---,pn in terms of the remaining coordinates
of the element x1,...,xn, 2h> •'••> Pn- The equation system
may therefore be thrown into the form
(3) p1 =J1 {xlf ..., xn, pm+i, ..., Pn), '••■>
Pm = Jm v^l> ■••» xn> Pm+l> •••> Pn)>
or into some equivalent form, obtained by replacing the
suffixes 1 , . . . , m by some m of the suffixes 1 , . . . , n. It is
obvious that, by differentiating the equations (3), we could
not obtain any equation connecting dXj, ..., dxn, and could
not therefore by the equation system assumed satisfy Pfaff's
equation.
We must therefore suppose that the equation system (2) is
such that at least one equation between x1, ..., xn alone can
be deduced from it. Suppose that exactly s of these equations
can be deduced; and suppose further that these have been
thrown into the forms
%n == fn (*u •••> ^n-s/f ■••> ^n-s+1 = Jn-s+l v*l» '"> xn-s)'
We now have
i = n—s t = g
p1dx1+... +pn dxn = 2 (Pi + 2 Pn-s+t %L*+t ) dxi 5
and therefore, if the equations (2) are to satisfy Pfaff's equa-
tion, we must have
* = s }>f
Pi+^Pn-s+tJfrLt= °> (*= i, ...,»-*);
for, by hypothesis, xx, ..., xn_s are unconnected.
0 2
196 PFAFFIAN SYSTEMS [152
We therefore conclude that every equation system satis-
fying PfafF's equation must include the system
xn — Jn 0*1 > •••' xn-s)> •••» xn-s+l —Jn-8+1 0*1 > •••> %n-s)>
t = s -\f t = s
2 J n-s + t ^ ofn_s+t
Pn-s+t ^Xn_g — °> '••' Pl^~ ^Pn-8+t~~^Z = °-
To these equations we may add a number of arbitrary
equations connecting xx, ..., xn_s, Pn-e+n •'•>Pni these equa-
tions, however, must be such that no equation of the form
/ 0*i> •••' xn-s) = 0
is deducible from them.
A set of equations satisfying PfafF's equation is called
a Pfajffian system. If the system contains m unconnected
equations it is said to be of order m, and we have proved
that m < n. When the number m is not specified it is to
be understood as being equal to n, and a Pfafiian system as
being of order n unless expressly stated to be of order m.
The equations of the system which do not involve p1, ...,pn
will be called the generating equations. There must be at
least one generating equation, and there cannot be more than
n ; there are, therefore, n classes of generating equations, if
we measure the class by the number of unconnected gener-
ating equations in the system.
§ 153. We now proceed to express in a convenient form the
conditions that n equations should form a Pfafiian system.
Let v be any function of the variables xx, ...,xn, plt ..., pn;
and let v denote the operator
t)v <) civ c) Dv <) Dv t)
+ ...+
tpt 2>% *pn *xn ^xx ^ lxn lpn '
then, u being any function of the variables,
i = n
^o / ^V 011 cv ou \
K^Pi^i *Xi*PiJ
We call the expression on the right the alternant of the
functions v and u, and we denote it by (v, u) ; we have
v . u = (v, u) = — (u, v) = — u . V.
The equation v — 0 will admit the infinitesimal trans-
formation
153] EQUATIONS IN INVOLUTION 197
if, and only if, the equation (u, v) = 0 is connected with
v = 0 ; that is, if the values of the variables, which satisfy
the second equation, also satisfy the first.
A set of functions ux , ..., um is said to be in involution
when the alternant of every pair vanishes.
So also a set of equations,
ux = 0, ..., um = 0,
is said to be in involution when for all values of the suffixes
(Uj, Us) = 0 is an equation connected with the given set.
An equation system in involution,
(1) u1 = 0,...,um = 0,
will therefore admit the m infinitesimal transformations
(2) ati^xt+t^-, p'i=Pi-t^-, (:;*"■•*)■
If v1 = 0, ..., vm = 0 is any given equation system such
that each of these equations is connected with
ux = 0, ..., um = 0,
and each of the equations u1= 0, ..., um= 0 is connected
with vx = 0, ..., vm = 0, we say that the two systems are
equivalent.
We must now prove that, if any equation system is in
involution, then any equivalent system is also in involution.
If v = 0 is connected with the system (1), it must admit
all the infinitesimal transformations which (1) admits; and
therefore (v> UJ = o, ..., (v, um) = 0
are equations each of which is connected with (1).
If then vx = 0, ..., vm = 0 is equivalent to (1) we know
that the equation (^ , u •) = 0 will be connected with
U1= 0,..., um = 0;
and therefore ux = 0, ..., um = 0 will admit the m infini-
tesimal transformations
<>V; ^V; ,4 — T ti
(3) 3$=*. + *-^ ti = Pi-t^> /*--l,...,»Y.
Now each of the equations v1= 0, ..., vm = 0 is connected
with ux = 0, ..., um = 0 ; and therefore each of these equations
admits the infinitesimal transformations (3) ; that is, the equa-
tion (viy vA = 0 is connected with ux = 0, ..., um = 0, and
198 HOMOGENEOUS EQUATION SYSTEM [153
therefore with ^ = 0, ..., vm = 0; that is, vx — 0, ..., vm = 0
are equations in involution.
If v15..., vTO is a set of functions of x1 , ..., xn, plt ...,pn, in
terms of which we can express Uj , . . . , uTO ; then, if ux , . . . , uTO
are unconnected, we can express v1,...,vm in terms of
ul,..., um; we say that two such systems of functions are
equivalent.
When we say that a function is homogeneous we shall
mean that it is homogeneous in p1, ..., pn; suppose that
ul} ..., um are each homogeneous functions, then, if vlt ..., vm
is an equivalent function system, v» will not in general be
a homogeneous function ; but, since there are m homo-
geneous functions, equivalent to i\, ..., vm, we shall say that
v, , ..., vm is a homogeneous function system. When each of
the functions v±, ..., vm is separately homogeneous, we shall
say that the homogeneous function system is in standard
form.
Similarly, if we say that the equation system
v1 = 0,..., vm= 0
is homogeneous, that will not mean that each separate equa-
tion is homogeneous, but only that an equivalent system can
be found, viz. u — 0 u = 0
"'l u5 •*•) "'TO '
each equation of which is homogeneous in px, ..., ptt.
It can be at once verified that the n unconnected equations
xn~fn\xl> •'•' xn-s) ~ "> •••> xn-s + l~Jn-s+l \xl> •••> xn-s) = ®>
t — s _ „ t = «
are in involution ; and that each of these equations is homo-
geneous ; we have, therefore, the following theorem : if m
equations form a Pfaffian system, it is possible to deduce from
them n unconnected homogeneous equations in involution.
The most important Pfaffian systems are those in which
m = n, and we see that n equations cannot form a Pfaffian
system unless they form a homogeneous equation system which
is in involution.
§ 154. We shall now prove the converse of this theorem,
viz. that a homogeneous equation system of order n in in-
volution forms a Pfaffian system.
154]
IN INVOLUTION
199
Suppose that in the system there are s generating equations,
viz. J1 (%! , . . . , xn) = 0, . . . , fs [xx , ..., xn) = 0 ;
and let the remaining (n—s) equations of the system be
thrown into the form
Ps+i — (Ps+l \Pl> •'•>Ps* xi> ••■> xn) = 0> •••>
Pn t* n \Pn • • • j Ps ' ^1 ' • • • ' %n) = ^ »
where the functions 4>s+1, • ••, <£M are homogeneous and of the
first degree.
We must first prove that the Jacobian determinantal
equation
— - — - 5 . • • —
(1)
*/.
= 0
dfCj ' ' <)Xg
is not connected with the generating equations ft — 0, . . . ,fs = 0.
Since (2>,+1-<fc,+i,/i) = 0, ..., (ps+1—<t>s+1, fs) = 0,
^s+i ^1 tyi
^a ty«
and therefore, if the equation (l) were satisfied for those values
of the variables which satisfy the generating equations, all
8-rowed determinants of the matrix
~bxx
would, when equated to zero, be equations connected with
the generating equations.
Proceeding thus, from the equations
(ps+2-<t>s+2>fi) = °> —>(Pn-4>n>fi) = °> (i= 1>->s)>
200
PFAFF'S EQUATION
[154
we should similarly see that all s-rowed determinants of the
matrix
ifi, . . . M
ixl
would also, when equated to zero, be connected with the
generating equations.
Now this is impossible ; for, were it true, it would mean
that, 03j, ..., xn being the coordinates of a point P on the
(n — s)- way locus
/i = 0, ...,/, = o,
and x1-\-dx1, ..., xn + dxn the coordinates of a consecutive
point P' on the (n—s+ l)-way locus
A = 0, ...,/,_! =0,
P' must also be on the (n — s)-way locus; and this is of course
not true, since the equations which define the locus are
unconnected.
The Jacobian determinantal equation is therefore uncon-
nected with the generating equations ; and we may therefore
throw the equations of the given homogeneous involution
system into the forms
xl — Ji \xs+l> '"■> xn) = ", ..., X8~J8 v^s+l' ■••> xn) = "»
Ps+l~ts+l \Pv •••jJP*j xs+l> •••> xn) = 0, ...,
Pn Jn\Pl> "'iPs' X8+l> ••'>xn) = "j
where /«+!,..., A are homogeneous of the first degree in
.Pi J • • • 5 .Ps '
By reason of the homogeneity of these functions we have
Js+j ~ J*Pi
(j = l,...,w-s),
and, since (ps+j-f8+j, xi~fi) = 0, we have
*Pi
we therefore conclude that
■— — =0;
<)x
'«+/
if
Pa+j + ^Pi^1- = °> U = *> ...,»-«)•
t>£C
s+j
155] GEOMETRICAL INTERPRETATION 201
From these (n — s) equations together with
xx—jx = 0, ...,xs—fs = 0,
we now at once deduce Pfaff's equation.
We have therefore proved that the necessary and sufficient
conditions that n unconnected equations should form a
Pfaffian system are that the equations should be homogeneous,
and in involution.
§ 155. We now know that xn_1 elements of space will
satisfy any assigned Pfaffian system of n equations between
the coordinates of the elements xx, ... xn, px, ... pn. If the
system contains only one generating equation, then the
elements consist of the points of an (n— l)-way locus in this
space together with the corresponding tangent planes to the
locus. If there are two generating equations fx (xx, ...,xn) = 0,
f2(x1, ...,^'rc) = 0 the elements consist of the points of this
(n— 2)-way locus together with the tangent planes which can
be drawn at each point of the locus ; there is not now, how-
ever, one definite plane at each point xx,...,xn, but an infinity
of tangent planes, viz.
M(^ + ^2)+...+(<-^(^ + ^) = o,
v i u ^ OXx OXxy n n/ v dXn dXnJ
where A : // is a variable parameter and x[, ...,x'n are the
current coordinates.
If there are three generating equations fx = 0, /2 = 0, /3 = 0
the elements will be formed by the points of this (n — 3)- way
locus together with the go2 of tangent planes, viz.
v 1 v OXx <SXX dXxJ
v n n/v *xn r*xn TixJ
and so on.
Each of these different classes of xn_1 elements satisfying
the Pfaffian equation
px dxx + ... +pn dxn = 0
will be denoted by the symbol Mn_x ; each will form a mani-
fold of united elements with (n— 1) ' degrees of freedom.'
Thus, when n = 2, that is, in two-dimensional space, the
elements are the points with the straight lines through the
points. The symbol Mx will now denote either an infinity of
202 EXTENDED DEFINITION OF THE [155
points on some curve together with the corresponding tangents
to the curve ; or a fixed point with the infinity of straight
lines through the point ; either of these infinities of elements
will satisfy the Pfaffian equation
pldx1+p2dx2 = 0.
In three-dimensional space there are co5 elements consisting
of points with the planes through them. The symbol M2 will
now denote one of three co2 sets of united elements, viz. (1)
the points of any surface with the corresponding tangent
planes ; (2) the infinity of points of any curve together with
an infinity of tangent planes passing through each point of
this curve ; (3) the co2 of planes passing through any fixed
point ; the elements of any one of these three sets will satisfy
the Pfaffian equation
px dxx +p.2 dx2 +p3 dx3 -- 0.
§ 156. We must now consider Lie's definition of an integral
of a partial differential equation of the first order ; and we
need only take the case where the equation is homogeneous,
and the dependent variable does not explicitly occur ; for
any partial differential equation of the first order can be
reduced to such a form (Forsyth, Differential Equations,
§ 209).
Let f{^x,...,xn, p1, ...,pn) = 0
be such an equation ; according to the usual definition
(f> (a?} , . . . , xn) = 0 is said to be an integral if, and only if,
/(
5,, ..., xn, r-^> •••>^-H = 0 is connected with <f> = 0.
1 n ox1 cxj
Stated geometrically, any surface — that is, any (n— l)-way
locus — is said to be an integral, if the coordinates of the
tangent plane, at any point, are connected with the coordinates
of the point by the equation
J \®\ s • • • ? ^n ' Pit ' • •> Pn) = ^'
Otherwise expressed, if we have any Mn_1} whose elements
satisfy the given equation, and which has only one generating
equation, then that generating equation is said to be an
integral of the given equation. Lie extends the notion of
an integral by defining it as the generating equations of any
Mn_x, which includes, as one of its Pfaffian system, the given
differential equation
156] INTEGRAL OF AN EQUATION 203
If then
/l v*'i> •••) ®n*Pli '">Pn) = ^j ■ '•>J?i v^iJ •••> xn^Pl'> '"iVw ~ "
is any homogeneous equation system in involution, such that
/= 0 is connected with fx = 0 fn = 0, the generating
equations of this system will be an integral, whatever the
number of these generating equations ; whereas, according
to the usual definition, they would only be an integral if the
number was one. By this extension of the definition of an
integral, it will be seen that more uniformity is introduced
into the theory of the transformations of partial differential
equations of the first order.
It should be noticed, however, that it is only special forms
of differential equations which can admit these new integrals.
If the equation
/ (x1 , . . . , xn , p1 , . . . , pn) = 0
has an integral of the form
■'&■■
xn — Jn (*l> •••' xn-s)> '••> xn-s+l — Jn-s + 1 V^'l' •••» xn-s)>
the equation must be satisfied for all values of
Xj, ..., Xn_8, 2^n-s+l' '••iPn>
when we substitute in it for xn) ..., xn_8+1 the respective
functions /„,..., fn-g+i> an(^ f°r Pk (where k may have any
value from 1 to (n — a)), the sum
-2^
S+J
Now to satisfy these equations it would in general be
necessary that the functions fn) ...,/n-«+i should satisfy a
number of partial differential equations, and, this number
being generally greater than s, the equations for/M, ...,/n_s+1
would not usually be consistent.
If, however, the given differential equation is the linear one,
Plp1+...+PnX>n - 0,
where P1 , . . . , Pn are functions of xx , . . ., xn , it will admit these
extended integrals. To prove this, let
be the integral equations of any characteristic curve defined by
(Jjjb-t \XfJbn (X>Jb~,
~p~ = p = ••• p 5
•* 1 ■* 2 -*■ n
204 ON FINDING THE COMPLETE [156
then
P^3+...+p J^ = 0, (k=l,...,n-l).
1<)X1 <>xn
From these conditions it follows that
P1p1+...+Pnpn = 0, %!-«!= 0, ..., «„_!-«„_! =0
are n homogeneous equations in involution ; and therefore
ux — a, = 0, ..., un_1 — an_1 = 0 are generating equations of
a Pfaffian system, which includes the given linear equation ;
it follows that an integral of
P1p1+...+Pnpn= 0
will be % = a19 ..., un_x = an_lt
where als ...,an_1 are any constants.
§ 157. In order to find the complete integral of
J \XH •••saV> P\i "•)Pn) = "j
we must find (n— 1) other unconnected homogeneous equations,
forming with / = 0 a Pfaffian system ; the generating equations
of this system will be (in Lie's sense) a complete integral if they
involve (n — 1 ) effective arbitrary constants.
Suppose that
/l (*i 5 •••» xn> -Pi' ■••) Pn> = ^s •••> J m \xl> "m^bi .Pi j '••> Pn) = "
are m given homogeneous equations in involution ; we can
throw these equations into such a form that some m of the
variables xlt ...,xn,p1, ...,pn will be given in terms of the
remaining ones.
Letaj19 ...,xm_s> plt ...,ps be given by
xi~Ji\xm-s+l> •••5iCn> Ps+n •••>Pw = "» (* = *s • • • > ^ — s)
Pj~$j \xm-s+l> •••>®n> Ps+l> •"■>Pn) = ^5 w = *> •••> s)'
These equations are still in involution ; but in any such
equation as (xi—fi, Pj — $•) = 0 the variables xx, ...,xm_s,
Pj, . ..,£>s do not occur at all ; and it therefore follows that the
above alternant, if it vanishes at all, must do so identically,
and not by virtue of any equation system ; the homogeneous
function system
xl~fl' '•■ > xrn-s~Jm-&> Pl~ $1' •••» Ps~<rs
must therefore be a system in involution.
158]
INTEGRAL OF AN EQUATION
205
If then we are given m equations in involution, and require
the remaining (n — m) equations forming with them a homo-
geneous Pfaffian system, we can reduce the problem to the
following : given m homogeneous functions in involution, it
is required to find (n — ni) other homogeneous functions,
forming with the given functions a complete system in
involution.
We shall show how one homogeneous function of degree
zero may be obtained ; having found this we shall have
(m+1) homogeneous functions in involution, and may proceed
similarly till all the functions are obtained.
§ 158. Let ux, ...,um be the given homogeneous functions
in involution, then, u denoting the operator
^p17iX1 "" *pn*Xn da^dft '" *Vn*Pn'
we see that if v is any function of u15 ..., u
m
W = V— Mi +
+
*v>m m
(this result is of course true whether or not u±, ...,um are in
involution) ; the operator v is therefore connected with the
operators u1} ...,um.
Conversely if v is connected with u1, ..., um, that is, if
v = k1u1+ ...+X
n um>
where A15 ..., Am are any functions of xl,...,xn, p1,...,pn:
then all (m+ l)-rowed determinants of the matrix
du.
~i>Ux ^Uj
~bux
*P1
*Pn
t)iCx
*xn
^Mm
Mi '
oxx
*xn
~}>v
~bv
~bv
i)V
*P1 '
M '
\ — ' *
oxx
^xn
must vanish identically ; and therefore v must be a function
oi ux , . . . , um .
Again, if u and v are any two functions of
•^l j • • • j 3-n ' Pi » • • •' Pn
206 ON FINDING THE COMPLETE
we see that
[158
— (u, v) = ( ^— , v ) + (u, — )
*Pi y*Pi J W
and therefore, u and i> being the corresponding operators, the
alternant u v — vu which is equal to
?>Pi " *PiJ *xi ' ^ K~ hxi " *V *Pi
i
i = n
»2(^:(—))^.-2(^(«.«)):
^»
do?,-
¥i
It follows that the alternant of u and v is derived from the
function (u, v) by the rule which derived the operator u from
the function u.
It is for this reason that we called the function (u, v) the
alternant of the functions u and v ; and what we have proved
is expressed symbolically by
(w, v) = (u, v).
If then u and v are in involution the operators u and v are
commutative, and conversely.
§ 159. Let the operator p1 \-...+pnz — be denoted by
P ; we shall now prove that P is not connected with
, um. Suppose it were so connected, then every
u
i •
(m + l)-rowed determinant of the matrix
*P1 '
du2
*Pn '
*xn
*Pn '
oxx
0, .
• o,
Pit '
• Pn
would vanish identically.
159] INTEGRAL OF AN EQUATION 207
It follows that every m-rowed determinant of the matrix
dUj ~bux
*Pi
?>p
n
1)U
in
~bu
III
^Pl
*Pn
must vanish ; there must therefore be some function of the
form (p (ux, ..., um) which does not involve .Pi ,'..., pn. By
passing to an equivalent function system we may take this
function to be um, where um only involves xx, ..., xn.
Every (m + l)-rowed determinant now vanishes in the matrix
i>U1
*2>1
*Pn
0, .
. 0,
0, .
• o,
ix^ ' ~*xn
^ ' *xn
Pl> ■ ' Pn
Now um does not contain plt ...,pni so that every two-
rowed determinant of
7>u
in
t)U
m
^Xx
Pn • ' ' Pn
cannot vanish ; else would um be a mere constant, which is
contrary to the hypothesis that ux, ..., um are unconnected.
We must therefore conclude that every (m— l)-rowed
determinant of
<>ux
*Pi
<)UX
*Pn
i>U
m-i
*Pi
<)u
m-\
*pt
vanishes identically.
We now proceed as before, and passing to an equivalent
208
ON FINDING THE COMPLETE
[159
system to u1} ...,wm_i may assume that um_Y does not contain
px,...,pn\ and we thus see that either every (m — 2)-rowed
determinant of the matrix
c>ul
*Pi
*Pn
en,
m-2
^>U
m-2
lPi • Mi
vanishes identically ; or else every 3-rowed determinant of
du
m-l
~bu
m-l
'&X1
2>xx
Pi,
DxTi
<>X
n
Vn
vanishes identically.
Since um_1 and um are functions of xx ,
t//»
alone, we see,
as before, that the latter hypothesis is untenable ; proceeding
with the alternate hypothesis, we ultimately come to the
conclusion that our hypothesis of P being connected with
U-,
vi>
um is untenable.
§ 160. If u is a homogeneous function of degree s in the
variables^, ...,pn it can be at once verified that
(P, u) = (s— 1) u.
The problem of finding a homogeneous function of degree
zero, in involution with each of the m homogeneous functions
ul9 ...,um (themselves mutually in involution), and uncon-
nected with these functions, is therefore equivalent to that
of finding an integral of the complete system of (m+1)
unconnected equations
uj=0,...,umf=0, Pf=0,
which shall not be a mere function of ux, ..., um.
There are (2n — m— 1) common integrals of
uj= 0, ...,um/ = 0, P/=0;
if any one of the functions ulf ...,um is of zero degree then
it will be an integral. There must, however, be at least
161] INTEGRAL OF AN EQUATION 209
(2 n — m — 1 — m) common integrals unconnected with uv . . .,um ;
and, as m is less than n, we can find at least one integral of
zero degree unconnected with ux , . . . , um .
We now see how the complete integral of a given equation
/ (&! , . . . , xn , px , . . . , 2\) = 0
is to be obtained.
We may write the equation in such a form as to give one
of the variables in terms of the others; say in one of the forms
(1) xx = 0! (#2s ..., xn, px, • ..,pn),
or, (2) px = fafa, ...,xn, p2, ...,pn).
We must then find, if we take the first form, a homogeneous
function of zero degree in involution with x1 — (})li and uncon-
nected with it ; knowing then two homogeneous functions in
involution, we find a third homogeneous function in involution
with these two, unconnected with them, and of zero degree;
proceeding thus, we finally obtain n unconnected functions in
involution, one of which is xl~<^1.
If we equate each of these functions, except xl — <l>1, to
arbitrary constants, and xx — $x to zero, we shall have a
Pfaffian system of equations which will include the given
equation, and will involve (n— 1) arbitrary constants; the
generating equations of this system will be a complete integral.
If we had taken the second form we should have proceeded
similarly.
§ 161. An equation of the form/C^, ...,xn) = 0 would not
ordinarily be called a differential equation ; but considering
Lie's extension of the definition of an integral it should be
regarded as a particular form of the differential equation.
If/^j, ..., xn) = 0 is one of this class of differential equations,
then any other unconnected equations of the form
J [X^f .. ., Xn) = 0, ■••>/n_i (2-1 j •••> Xn) = U
will with / = 0 form a Pfaffian system : any point on the
locus/ = 0 will be an integral of the equation/ = 0. These
integrals are also complete integrals ; for the coordinates of
any point on the locus / = 0 will involve (n — 1) arbitrary
constants.
If the assigned differential equation is of the form
(v Pi /l vhi ■••> xn) + ••• +Pnfn \®l> •••> xn) = ">
CAMPBELL P
210
EXAMPLES
[161
we could also have 'point' integrals, the equations which
define each point generating a Mn_1 ; these points, however,
will in general be isolated points satisfying the equations
f = 0, ...,fn = 0, and will not therefore be complete integrals.
Suppose that the equations /: = 0, ...,/„ = 0 are equivalent
to a smaller number of equations, say
<Pj (A'-, , . . ., Xn) = U, . . . , <pm [X^ , • . . , Xn) = U ,
we should have an (n — m)-wa,y locus in space, any point of
which would be an integral of the given equation (1) ; these
integrals, however, would not be complete, since they would
only involve (n — m) arbitrary constants.
§ 162. Example. Consider the equation
•^1*^2 Ps == X'dPlP'2'
of which a complete integral is
a\ x\ + a\ x\ + ctj a2x'l+l = 0.
The corresponding Pfaffian system is
Pi Pz P3
Cv-i Jb-t
Cvo i&9
Ct--i t*o «//q
5 a\ x\ + a| scf + ax a2 x\ + 1 = 0,
which may be thrown into the form
Pi
Pi
+ a\ = 0 ,
+ a% = 0, x1x2pl-xlp12)2 = 0
x2 (pj xx +p2 x2 +p3 xz)
These equations define an oo2 of M2s, each of which consists
of points on a surface together with their corresponding
tangent planes.
We shall now try whether the given equation can be
satisfied by an oo2 of M2's, each of which consists of points
on a curve together with the infinity of tangent planes which
can be drawn at each point of this curve.
Let the generating equations be
XZ =/(-ri)> X2 = <t>(Xl),
then the third Pfaffian equation must be
Pi +P2 *'(Sl) +Pi f'(xi) = 0,
163] EXAMPLES 211
where /' denotes the differential coefficient of / with respect
to its argument.
\i xxx2p\ — x\pxp2 = 0 is to be connected with this Pfaffian
system, we must have
^4>Pl + f2¥pl+f2fp,P, = ^
for all values of oc1,p2,p3 ; and therefore we must have (p = 0
and/' = 0 for all values of the argument xv
From the third Pfaffian equation we now conclude that
p1 = 0 ; and therefore
x2 = 0, x3 = constant, px = 0
will be an co of M2's satisfying the given differential equation ;
we do not, however, obtain an go2 of the required class of M2's.
Example. Find the complete integrals of
p1x1-\- ... -\-pn xn = 0
which are straight lines.
§ 163. As an example of an equation having no integral
which is a curve, take
pl+p22 + 2p1p3x1 + 2p2p3x2+2plx1x2 = 0
(Forsyth, Differential Equations, § 202, Ex. 1).
If the Pfaffian system
«b =/ («a)> xz = <t> Wi Pi +Pz $ («i) +P3 /' (xi) = °
were to satisfy this equation, we should have
4>'2+l=0, P-*xx(f '-</>) = <>, ^'(Z'-^ + ^O;
and, as these equations are inconsistent, we conclude that the
given equation has no integral of the required form.
In order to obtain examples of equations having integrals
in Lie's extended sense, it is only necessary to write down
any equations
/ 1 \xl > • • • J xn) = ^s • * • ' Js \xl ' ' " ' ' xn) = '
involving (n— 1) effective arbitrary constants, and then to
complete the Pfaffian system.
Let
Js+l \®1> •••' ^n'Pl) •"■>Pn) = "» "•' In \xl> "•>xn>Pl> •">Pn) = "
be the remaining equations of the system ; if we eliminate
P 2
212
EXAMPLES
[163
the arbitrary constants from the system we shall have a single
equation between x1, ..., xn, plt ...,j)n', the complete integral
of this equation will be
J\ v^'ij •••j xn) = ®> •••»/« (*^l> •••' %n) = ®'
Example. Take the equations
x3 = axx + b, xi = a2x2 + c
where a, b, c are arbitrary constants ; the other two Pfaffian
equations will be
and therefore p{ p± +p2 pi = 0 is an equation with the complete
integral
CHAPTER XV
COMPLETE SYSTEMS OF HOMOGENEOUS
FUNCTIONS
§ 164. Let Uj,...,um be m unconnected homogeneous
functions of x1} ..., xn, plt ...,pn- If we form the alternant
of any two of these functions u$ and u- we obtain the
homogeneous function (u^ v>J) ; if (ui} Uj) is unconnected with
, um we add it to this system and have thus (m+1)
u
u ■ • • s wm
unconnected homogeneous functions. Proceeding thus, since
there cannot be more than 2 n unconnected homogeneous
functions, we must ultimately obtain what we call a complete
system of homogeneous functions ; that is, a system of functions
homogeneous in px, ...,pn, and such that the alternant of any
two functions of the system is connected with the functions
of the system.
Let us now take ux, ...,um to be a complete homogeneous
function system, so that we have
A = 1, ...,ra\
(ui,uj) = wij(u1,...,um) (. % jm;-
The functions w^ of the arguments ux , . . . , um are called
the structure functions of the complete system; and, since
(uit Uj) + (uj , u{) = 0, we must have Wg+Wji = 0.
If vlt ..., vm is a system of functions equivalent to ult ..., um
(that is, if for all values of the suffix i, v{ can be expressed
in terms of ux, ..., um, and u{ in terms of vlt...,vm), then,
though v1,...,vm may not each separately be homogeneous
functions, we call vlt...,vm& homogeneous function system.
If then we are given a system of functions vlt ..., vm of the
variables xlt .,.,#„, £>i, ..•>#»> now are we to know whether
or not the system is a homogeneous one "?
Denoting by P the operator
we shall prove that the necessary and sufficient conditions
214 HOMOGENEOUS FUNCTION SYSTEM [164
that the system may be homogeneous are that Pvx, ...,Pvm
should each be connected with vx, ..., vm, that is, each be
expressible in terms of vx, ...,vm.
Firstly, the conditions are necessary; for if ux, ...,um are
on functions homogeneous in px, ...,pn and respectively of
degrees 8-,,...,8m, and forming a system equivalent to
Jmi
V
,., vm , then P^ is connected with ux , ...,um, Pux, ...,Pum.
Now Pu; is equal to Sj u- , and therefore Pi\ is a function of
a
i •
,,um, and so also a function of vx
,vm; we thus see
that the conditions are necessary.
Secondly, these conditions are sufficient ; for suppose that
Pvi = A>i> • •>%«)> (* = l,---,m);
then if fl3 ...,fm are each identically equal to zero, vx, ...,vm
will be homogeneous functions of zero degree. If on the
other hand these functions do not vanish identically, we can
find (m— 1) unconnected functions of vx, ...,vm such that they
are each annihilated by
f*5* +
+/■
m
<)v,
m
and therefore by P.
Let these functions be ux, ...,um_x; they will be homo-
geneous functions of degree zero ; we can then find one other
function of vx, ...,vm say um, unconnected with ux, ...,um_1,
and satisfying the equation
f *um
+ ...+/,
t>U„
7)1
"TO"
and therefore satisfying the equation Pum = u
The function um is therefore homogeneous of degree unity ;
and, as the system ux, ...,um is equivalent to vx
vm, we
conclude that the necessary conditions are also sufficient.
§ 165. If u1,...,um are m unconnected functions of
xx, ...,xn, px, ...,pn, which mayor may not be a homogeneous
system, we say that the system is complete if the alternant
of any two of the functions is connected with u-, u,
u
m*
If then we form the alternant of/ (ux , . . . , um) and <fi (ux, ..., um)
(where / and $ are any two functional symbols) we see that
this alternant is connected with ux , . . . , um , if ux , . . . , um are
the functions of a complete system. It at once follows that
um, the
V
vm being any system equivalent to ux,
one system is complete, if the other is complete.
m'
166] COMPLETE HOMOGENEOUS SYSTEM 215
We can now give a general definition of a complete homo-
geneous function system, as a system of m unconnected
functions ux, ...,um such that
(UfiUj) = w{j (u1,...,um), i - i} ...>m.
Pu{ = wi(u1,..., um), J = 1> • • -, *»'
The functions w^t ...,W{, ... are the structure functions
• of the system ; we can pass to any equivalent system vl, ...,vm,
and in so doing we should change the form of the structure
functions. Thus when we pass to an equivalent system in
which vx, ...,vm_1 are homogeneous of degree zero, and vm
homogeneous, either of degree zero or of degree unity, we
have w11...swm_1 each zero, and wm either zero or unity.
The main problem to be considered in this chapter is how
to pass to a system equivalent to ux, ...,iim in which the
structure functions may have the simplest possible form.
If each function u1? ...,um is homogeneous and of degree
zero, then f (v^, ...,um) is homogeneous and of degree zero;
and therefore every equivalent system has all its functions
of degree zero. If such a system is complete, we shall now
prove that it is in involution.
Since (u^uA is by hypothesis a function of ulf ...,um,
it is homogeneous and of degree zero ; but w^ and u- are each
homogeneous of degree zero, and therefore their alternant
is homogeneous and of degree minus unity. The only way
of reconciling these two facts is by supposing that (ui, ...,uA
is identically zero ; that is, the system must be in involution.
§ 166. We shall, as in § 153, denote by ut the operator
oi)1ox1 '" *Pn*xn ^>x1op1 '" *anZpn'
and by (ui , u .•) the alternant of u^ and u >> . We have proved
that this operator is derived from the alternant of the
functions u^ and Ui by the rule which derived the operator
u~i from the function u$.
We have also proved (§ 159) that the operators ul9 ...,um
and P are unconnected. If we form the alternant of P and u$
we get
Pj J *XjS * Pi ^OXjO pj
** dp- V l/ OX; j6md OXj x op-
216
AN IMPORTANT IDENTITY
[166
that is, the alternant of P and u{ is derived from the function
(P — 1) u{ by the rule which derived u{ from u{.
If then uls ...,um are functions forming a complete system,
the operators ult ...,um form a complete system; and if
um form a complete homogeneous system, ut, ..., um, P
u
i'
will be a complete system of (m+ 1) unconnected operators.
The operators u19 ..., um form a complete sub-system of
operators within the system ux, ...,nmP; and the alternants
(P, Uj ),..., (P, um) are each connected with u1, ...,um. From
these facts we conclude that the complete system of equations
(u1,f) = 0,...,(um,f) = 0,
admits the infinitesimal transformation
Pi^Pt+tPh (i = i, ...,»);
and therefore, i// is any function annihilated byv,li...,um,
Pf will also be annihilated by these operators.
§ 167. We shall now prove an important identity which
will immediately be required.
If u, v, w are any three functions of the variables
X-^ , . , . , xn , Pi , . . . , pn .
then it will be proved that
(u, (v, w)) + (w, (u, v)) + (v, (w, u)) = 0.
Since (uPu) = (w, v)
it follows that the operator derived from
(1 ) (u, (v, w)) + (w, (u, v)) + (v, (w, u))
is (u, (v, wj) + (w, (u, v)) + (v, (w, u)).
Now by Jacobi's identity this operator vanishes identically
and therefore (1) must be a mere constant. We next prove
that this constant is zero.
If we notice that
(uv, w) = u (v, w) + v (u, w),
we may easily verify that
(u\ (v, w)) + (iv, (u2, v)) + (v, (w, u2))
= 2u [(u, (v, wj) + (w, (u, v)) + (v, (w, u))] ;
and we therefore conclude that the constant — 2u x some
constant.
Now u being any function whatever of the variable, this
168] THE POLAR SYSTEM 217
can only be true if the constants are zero ; and therefore
we see that
(u, (v, w)) + (w, (u, vj) + (v, (w, %)) = 0.
Another proof of this theorem is given in Forsyth, Differential
Equations, § 214.
Let now u15 ...,um be a complete system ; then v^, ...,um,
being unconnected, there must be (2n — m) unconnected
functions of the variables which will be annihilated by
«l, •••j wm. Let these functions be i\, ..., v2n_m ; we must
now have (ui} Vj) = 0 for all values of the suffixes.
From the identity
K', ty . vk)) + (vk , (u{ , Vj)) + (vj , (vk , u{)) = 0
we conclude that (u{ (Vj,Vj.)) = 0, and therefore ut (v.-, vh) = 0.
We therefore have the theorem : every alternant of v1 ,...,v0n_m
is annihilated by the operators ulf ...,um.
Now every function annihilated by these operators will be
connected with v1} ..., v2n_m ; and therefore every alternant of
vx, ...,v2n_m is connected with this given set of functions;
that is, vlt ..., V-zn-m is itself a complete function system.
The m unconnected functions u1,...,um are annihilated
by each of the (2n — m) operators vx, ...,v2n_m, so that the
two systems are reciprocally related, and each is said to be
the polar of the other.
If u1,...,um is a complete homogeneous system its polar
system is also homogeneous. For u1,...,um is homogeneous,
and Vj is annihilated by u15 ...,um; therefore (by § 166) Pv{
is also annihilated by u15 ...,um; Pv^ must therefore be a
function of vx, ..., v.2n_m ; that is, v1} ..., v.2n_m is a homogeneous
function system.
Suppose that we are given a system ux, ...,um such that
(ui> uj) — wij (ui5 •••Jum)' A = 1, ...,
PUj=z Wj K, ...,UJ, V
/% = 1, ...,m\
\j = 1, ...,mJ
any function whatever of ult ..., um will be a function of the
system, but we regard u15 ...,um as the fundamental set of
functions of the system once we have chosen them ; if we
were to change to an equivalent set of fundamental functions
we should have to change the structure functions.
§ 168. It must now be proved that the functions which
are common to a system and its polar system — that is, the
functions which are connected with ulf ...,um and also with
218
THE ABELIAN SUB-SYSTEM
[168
vx, -..,v2n_m — will themselves form a homogeneous system in
involution.
Let ux, ..., um+q be a complete homogeneous system; by
properly choosing the fundamental functions of the system we
may suppose that wm+x, ..., um+ are the functions of the
given system which belong also to the polar system.
Since um+x, ••■,um+q are each annihilated by ul9 ..•,um+q
they are functions in involution ; and, since both the given
system and its polar system are homogeneous,
Pu
m + 1 '
P
U
m+q
'm + q
must be functions common to the two systems, and therefore
must be functions of um+1, -..,um+( ; that is, wm+x, ...,u„
is itself a homogeneous system.
We call this homogeneous sub-system of ult -..,um+ its
Abelian sub-system : if the Abelian sub-system coincides
with the polar system, we say that the given system is
a satisfied one.
If a system is satisfied its polar system is then a system
in involution; conversely, if a system is in involution, its
polar system is satisfied; for, if vx, -..,v2n_m is a system in
involution, all of these functions must also be contained in the
polar system ux, ..., um, which is therefore satisfied.
§ 169. Let ux,...,um be a complete homogeneous system
which is not satisfied ; its polar system is, we know, a homo-
geneous one ; but all the functions vXi ..., v2n_m cannot be of
zero degree, else would the polar system be in involution, and
ui,---,wm a satisfied system. The polar system can then
be thrown into such a form that vx is of degree unity, and
v,
2'
5 v2n-m
each of zero degree ; and it can therefore be
thrown into such a form that each of its fundamental set of
functions is of degree unity; for vx,vxv2, ...,vxv2n_m would
be (2n — m) unconnected functions of the polar system, each
of degree unity.
Since ux, ...,um is not satisfied, not all of the functions
vx, ..., v2n_m of the polar system can be connected with
uu •••■>wm' We may therefore suppose that vx is not so con-
nected ; and, as it is a homogeneous function of degree unity
in involution with ux, ..., um, we see that
«i,
• 5 um ' vi
is a complete homogeneous function system of order (ra-f- 1).
Every unsatisfied system is therefore contained, as a sub-
170] SYSTEMS OF THE SAME STRUCTURE 219
system, within another complete homogeneous system whose
order is greater by unity than that of the given system.
We thus see that we can continue to add new functions
to a given system, till it will finally be contained as a sub-
system, within a satisfied system.
§ 170. If we have two complete systems ux, ...,um and
vls ...,vm with the same structure functions ; that is, if
(U{, Uj) = wtj (ux, ..., um), Pu{ = Wi (ux, ..., uj,
(vit Vj) = w{. (v13 ..., Vm), PVi = Wi (i/ls ..., vm),
then, if one system is satisfied, so is the other.
To prove this consider the linear operator
wix (ux, ...,uj — + ... +wim (ux, ...,uj — ,
which we call the contracted operator of Uj . Let / (ux , . . . , um)
be any function of ux , . . . , um ; then, since
^ f ~> f
»,'/K um) = (ui> Ul) >~7 + ••• + K. um) ^~ >
° ui c am
we see that the contracted operator of ui has the same effect
on any function of ux , . . . , um as the operator u^ itself.
The contracted operator of P is
wx(ux,...,um)^— +...+wm(ult...,um)
m
Zux -»v~i»"-»-«/jtti
The Abelian sub-system of ux, ..., um consists of the
functions annihilated by the contracted operators of ux, ...,um.
If ux. ..., um is a satisfied system, every function annihilated
by ux, ...,um is also annihilated by the contracted operators ;
and therefore there are (2w-m) functions of ux, ...,um which
are annihilated by the contracted operators. Since the con-
tracted operators of vx, ...,vm are of exactly the same form
invx, ...,vm that the contracted operators of ux, ...,um are in
ttj,,..^,,, it follows that there are (2,w-m) unconnected
functions of vx, ...,vm annihilated by the contracted operators
of vx , . . . , vm ; and therefore vx, ...,vmis also a satisfied system.
If ux, ...,um is an unsatisfied system, we have proved that
a homogeneous function um+x can be added to it, such that
um+i *s °f degree unity, and in involution with ux, ...,um.
If then we have two systems ux, ...,um and vx, ...,vm, with
the same structure functions, we can add um+x to the first,
and vm+x to the second, in such a way that ux, ..., um+x and
220 COMPLETE SYSTEMS IN [170
v1} ...,vm+1 will still remain homogeneous function systems
of like structure.
We thus see that if we are given two systems ulf ...,um
and v1)...,vmi of like structure, we can add functions to
each, in such a way that the new systems become satisfied
simultaneously, and have, when both satisfied, still the same
structure.
§ 171. We must now show how a complete homogeneous
system is to be reduced to its simplest form.
We first find the Abelian sub-system of the given system
ux, ..., um, um+1, ..., um+ ; to find this it is only necessary
to form the contracted operators of u15 •..■>um+q, and then to
find the functions of u±, ...,um+q which these annihilate.
We may now suppose that the fundamental functions have
been so chosen that um+1J ...,um+_ is this Abelian sub-
system ; and we further suppose that each of the functions
ui> •'■>um+q are giyen m homogeneous form, so that u^ is of
degree si, in the variables px, ...,pn.
Since the contracted operator of u^ is
(U*' Ul>7^r+'~+ (Ui>U™+q) ^ >
owl Cllm+q
we see that the contracted operators of um+1, ...,um+ vanish
identically.
The contracted operator of u • , where j does not exceed m, is
J- Jib
and these contracted operators of u1,...,um cannot be con-
nected. For if they were connected, they would form a
complete system of operators in ux , . . ., um, and would therefore
have at least one common integral which would be a function
of Uj, ..., um. Now this integral, being a function annihilated
by uv ...,um+q, would be an Abelian function of the group,
which would be contrary to our hypothesis that um+1, ...,um+
are the only unconnected Abelian functions in the system.
The contracted operator of P is
and we have (as proved for the more general case in § 159),
(1) Pui-uiP=(si-l)ui.
172] THEIR SIMPLEST FORMS 221
We have proved that we may take the functions of the
system in such a form that they are either all homogeneous
of degree zero, or all but one of degree zero, and that one
of degree unity.
In the first case the functions are all in involution and the
system cannot be thrown into any simpler form.
In the second case the function of degree unity may be
an Abelian function, or it may be a non-Abelian function
of the system.
We consider these alternatives ; and we first suppose that
the Abelian function um+1 is of degree unity, and that
u1,...,um,um+2,...,um+q are each of degree zero.
§ 172. Each of the alternants (u^u^, ...,(u1,um) will now
be of degree minus unity, and therefore
will each be homogeneous functions of degree zero ; and, as
they are functions of uv ...,um+ , all of which except um+1
are of zero degree, we conclude that they are functions of
It now follows that some function of
11-^, ..., 1lm, Um + 2, ..., Um+q
can be found, say f(ult ...,iim,um+2,...,um+q), such that
um+l'ulf = * '■>
and therefore (since u1wm+1 — 0) um+1 f will be a function
ofu1,...,um+q, of degree unity in pt, ...,pm, and such that
Since um+1f cannot be an Abelian function of the system
(else would it be in involution with u15 and annihilated by uj,
we may therefore take the functions of the fundamental
system in such a form that u2 and also wm+1 are of unit
degree, whilst all the other functions are of degree zero ;
(%1}u2) = 1, and um+1, ...,um + q are Abelian functions.
Since {ux,u2) = l,Uj and u2 will be permutable, and there-
fore the contracted operators of ux and u2 will also be per-
mutable. There are therefore (m + q — 2) unconnected func-
tions of ux, ...,um+q annihilated by ua and u2 ; and, from the
formula (1) of § 171, we see that if f (uv ...,um + q) is one
such function Pf (ult ...,um+q) will be another such. These
functions therefore form a complete homogeneous function
222 COMPLETE SYSTEMS IN [172
system in themselves; and, since (ux,u2) — 1, each one of
these functions must be unconnected with ux and u.2.
It follows from the above discussion that we may take the
fundamental functions of the system in such a form that
ux and u2 are in involution with u3, ■■•<wm+ ; that
Um+1> '••>Um + q
are Abelian functions, and (ur,u) = 1 ; and further that u2
and Um+1 are each of degree unity, whilst the other functions
are of degree zero.
Since u3, „.,uffi+. is now in itself a complete homogeneous
function system, we may treat it in a similar manner, and
• thus reduce the function system to the form
Ux, Vj, l^ j ^2' •"> ^s' ^'s> ^s + l> •••' ^8 + q>
where ux, ...,us,vs+2, ...,vs+q are each homogeneous of zero
degree, and vx, -..,vs+1 are each homogeneous of degree unity;
and where further
(u^vj = (u2,v2) =...= (us,vs) = 1,
all other alternants of the system vanishing identically.
If instead of the functions vs+1, ...,vs+ , we take the
Abelian functions vs+1,vs+1vs+2, ...,vs+1vs+q, we obtain the
normal form. In this all the functions ux, ...,us are of
degree zero, all the functions vx, ...,vs + q are of degree unity,
and
(A) (Uj , vx) = {u2 ,v2)=...= (us , vs) = 1 ,
while all the other alternants of the system vanish identically.
§ 173. We next take the case where all the Abelian
functions are of degree zero, and we take ux to be of degree
unity, whilst all other functions of the system are of zero
degree.
Since (w^w^,..., («i *<*„,)
are each homogeneous functions of degree zero, they must be
functions of m2,...,ww+„ only; and we can therefore find
a homogeneous function of degree zero, say /(it2, ...,wwl+ ),
such that — /• ,
ux.J = 1.
We now see as in the last article that we may take the
functions of the system to be
^1)^2)^3) '•-ium + l' •••>um + q>
175] THEIR SIMPLEST FORMS 223
where (ux,u2) = 1 , and all the other functions are in involution
with these two, and form in themselves a complete homo-
geneous function system.
The system u?j, ...,um + q cannot have all its functions of
degree zero, else would these functions all be Abelian within
the system ult -••,ul)l+q, which is contrary to the hypothesis
that there were only q such functions.
We may therefore, since the Abelian functions are each of
degree zero, take u3 to be of degree unity.
We then, as before, reduce this system to the normal form
(B) ^i,^,^,^, ...,V>8,V8,Vg+1,...)Vg+q,
where ul,...,us are homogeneous of degree unity, and
vx, ...,vs+q homogeneous of degree zero, and where
K^i) = K>«2) =...= (u>8,v8) = 1,
whilst all the other alternants vanish identically.
Every complete homogeneous system is therefore such that
all its functions are of degree zero, and therefore all its
alternants vanish identically ; or it is equivalent to one of the
two forms (A), or (B).
§ 174. It is important to notice that, in bringing u1, ...,um
to normal form, we replace these functions by an equivalent
system of fundamental functions
and to find the forms of the functions fv ...,fm we did not
make use of the operators ul9 ...,um themselves, but only of
the contracted forms of these operators, viz.
If therefore ux, ..., um and vx, ..., vm are two complete homo-
geneous systems of like structure, and, if
/l V^u '••>um)> •••'Jm \W\i •••ium)
is a system equivalent to uly ...,um and in normal form, then
/l v^i? •••> vi,J> •••>fm (vi> •••' vm)
will be a function system equivalent to v1,...,vm, and will
be in normal form.
§ 175. We can now prove that a complete homogeneous
system, which contains Abelian functions, is contained as
224 COMPLETE SYSTEMS [175
a sub-system within a larger system, not containing any
Abelian functions.
We take the system in normal form (A)
Ui , . . . , U8 , Vj , ...,Vs + q,
where vlr ...,vs+q are each of degree unity.
The functions u1} ..., us, v8)v8+2, ...,vg+q now form a system
complete in itself; if we form the system polar to this it
must contain v8+1 ; but in the polar system v8+1 cannot be an
Abelian function, since it is not a function of the system
We can therefore find within the dual system a homogeneous
function of degree zero, say us+1, such that
We now have the homogeneous system
ult ...,il8+1, v1,...,vs+q,
which is of normal form but only contains C^ — 1) Abelian
functions. Proceeding similarly, we finally obtain a system
of (2 s + 2 q) homogeneous functions
Ulf ...,Us + q, Vlt ..., V&+q,
such that
(ux, wj = (u2, v2) =...= (us+q, vs+q) = 1,
and all other alternants vanish identically; u1} ...,u8+q are
each homogeneous of degree unity; V1}...9v8+q are each
homogeneous of zero degree ; and there are in the system
no Abelian functions ; that is. no functions in involution with
all functions of the system.
We should obtain the same results had we taken systems
of either of the normal forms
u^, ..., us, v^, ...,vg+q,
where v1} ...,vs+q are each functions of degree zero ; or
^l ' • • • » m '
where vlt ...,vm are all of degree zero, and therefore all in
involution.
§ 176. In a satisfied system, since the polar system is now
the Abelian sub-system, q = 2n — 2s — q, and therefore
2s + 2q = 2n;
if then we apply this reasoning to a satisfied system we see
176] IN NORMAL FORM 225
that it is contained in a system of order 2n, which has no
Abelian functions.
As we have proved that every complete system is contained
as a sub-system within a satisfied system, we see that every
system is a sub-system within a homogeneous system of
order 2n.
If Uj, ...,um and vlt ...,vm are two complete homogeneous
systems of the same structure, we can then take, as a funda-
mental set of functions of the first, a system
and as the fundamental functions of the second
and we can add functions to each of these systems, till finally
we have two function systems, of order 2 n, which will be
in normal form, will contain no Abelian functions, and will
be of the same structure, with fi(uv ...,wm) corresponding
to j i \vx, ..., vm).
CAMPBELL
n
CHAPTER XVI
CONTACT TRANSFORMATIONS
§ 177. We know (§ 154) that if Xx, ..., Xn are functions of
x x , . . . , xn , px , . . . , ^?w ,
homogeneous and of zero degree in px, ...,pn, the necessary
and sufficient conditions, in order that
may be a Pfaffian system of equations, for all values of the
constants ax, ..., an, are that Xx, ..., Xn should be unconnected
functions in involution. It follows that pxdxx+ ...+pndx.
will be a sum of multiples of dXv ..., dXn if, and only if,
Xx, ..., Xn are unconnected functions, in involution, and homo-
geneous in pXi ..., pn of zero degree.
If then we know n unconnected functions Xlf .,., Xn satis-
fying these conditions, n other functions Px, ..., Pn of the
variables xx, ..., xn, px, ...,pn can be found such that
Px dXx + . . . + Pn dXn = pxdxx + ... +pn dxn.
Let us now seek the conditions in order that
^i ~ %i> Pi = -Pi. (i=l,...,n),
where Xx, ..., Xn, Px, ..., Pn are unknown functions of
xx , . . . , xn , px , . , . , ^>n ,
may lead to the equation
i = n i = n
^p'idx't =^PidXi.
Consider the Pfaffian equation
i = n i = n
2 Pi d®i - 2 Pi dx\ = °
in the 4n unconnected variables
*^l> •••) ^n > .Pi» •••> i?«) ^ij •••> ^nj ^u •••> ^n*
177] CONTACT TRANSFORMATIONS 227
The necessary and sufficient conditions that the 2 n equations
(1) a£-X< = 0, Pi-Pi = 0, (i = l,...,n)
should satisfy it are the three following.
Firstly, the equations must be unconnected ; this condition
is evidently satisfied since x'x, ...,x'n, p'x, ...,p'n are unconnected.
Secondly, the equations
xfi-xi=°, (i = l, ...,»)
must be homogeneous of zero degree in px, ..., pn, p[, ..., p'n ;
and therefore Xx, ..., Xn must each be homogeneous of zero
degree in p1,...,pn. Similarly we see that jP15 ..., Pn must
each be homogeneous of the first degree in. px, ..., pn.
Thirdly, the equations must be in involution. It is easily
seen that the following identities hold for all forms of the
unknown functions Xx, ..., Xn, Pv ..., Pn, viz.
(x'i-Xi,x'k-Xk)=(Xi,Xk),
Vi-x* p'jc-Pu) = («4. p',) + (Xit pk) = (Xi, pk) if i * \
{x'i-Xi, p'i-Pi) = (aft,M + (X^ Pd=-l + (X{, P,),
(Pi-Pi,pk-Pk) = (Pi,Pk).
If then the given equations are in involution, we must have,
for all values of xx, ..., xn, ijx, ...,pn, x'x, ..., afn, p'x, ...,p'n
satisfying the equations (1),
(X^ Xk) = 0, (Xi, Pk) = 0ifi* k, (Xi, P£ = 1, (P{, Pk) = 0.
Now from the given equations (l) no equation connecting
xx, ..., xn, px, ..., pn can be deduced ; and therefore the given
equations cannot be in involution, unless we have identically
(X^ Xk) = 0, (X^ Pk) = 0 if ijb k, (Xi, Pi) = 1, (Pi, Pk) = 0.
We therefore have the following important theorem :
^i = ^i, Pi-Pi, (i=l,...,n)
will then, and then only, lead to
i—n i=n
^PidXi-^PidXi;
that is, to the identity
i= n i = n
^PidXi=^Pidxi,
if Xi is homogeneous, and of zero degree in px, ...,pn, P^ is
homogeneous, and of the first degree in px, ..., pn, and
(X{, Xk) = 0, (Xiy Pk) = 0 if i* k, (X^ Pi) = 1, (P{, Pk) = 0.
Q 2
228 FUNCTIONS DEFINING A [177
It must now be proved that there cannot be any functional
connexion between Xx, ..., Xn, Px, ...,Pn.
§ 178. Suppose that it were possible to express Pn in the
form Pn= V (Xx ,..., Xn, Px,..., Pn-i)>
where V is some functional symbol ; then we should have
(Xn, V) = (Xn, Pn) = 1 ;
and, since Xn is in involution with Xl,...,Xn, Px, ...,Pn_x,
it must be in involution with V, and therefore (Xn, V) would
be equal both to zero and to unity.
There cannot then be any connexion between Xx, ... Xn,
Px , . . ., Pn involving any of the functions P](...,PB. Suppose
that there could be a functional connexion between Xx, ...,Xn
alone ; then, since the equations
JLX = ax , . . . , An = an
(where ax, ...,an are any constants) satisfy Pfaffs equation
px dxx + ... +pn dxn = 0,
we know from § 1 54 that the given equations must be uncon-
nected ; and this result is inconsistent with the hypothesis
of X1} ..., Xn being connected.
We conclude then that Xx, ...,Xn, Px, ...,Pn are entirely
unconnected ; and therefore
(1) x\ = X{, & = P{, (i = 1 , . . ., n)
will be a transformation scheme since by means of this equa-
tion system we can express each of the variables x1,...,xni
p±, ...,pn in terms of x'{, ..., x'n, p[, ...,p'n.
The transformation acheme (1) is said to be a homogeneous
contact transformation scheme, since it does not alter the
Pfaffian expression, but transforms
i—n i=n
^Pidxj into 2.PW^-
The scheme we are considering transforms elements in space
x1,...,xn into elements in space x[, ...,x'n; and, if two con-
secutive elements of the one space are united, the corresponding
elements of the other space will be united. The danger of
a geometrical misinterpretation must be guarded against :
thus, if A is a point in one space and a a plane through A,
the point and the plane together make up an element of that
space ; if B is a second point in the same space and (3 a plane
179]
CONTACT TRANSFORMATION
229
through it then we have a second element in the same space.
Let now A' be the point in the other space which corresponds
to the element A, a (not merely to the point A) and a the
plane through A' corresponding to the same element ; and let
B' and /3' have similar meanings with respect to B, /3. If B
lies on a it is not at all necessary that B' should lie on a.
If, however, B is contiguous to A, and /3 to a, then B, j3 is
a contiguous element to A, a; and, if B lies on a, they are
united elements ; we then see (the transformation scheme
between the elements being a contact one), that B' lies on a,
and A' on ft', and that B', ($' and A', a are united elements.
§ 179. It is important to notice that the contact transforma-
tion scheme is altogether known when we know the functions
X1,...,Xn. To prove this let the known functions, homo-
geneous, of zero degree in ply...,pn, and in involution, be
X1; ...,XW. We have proved that functions Px, ...,Pn must
exist such that
Pj dXr + . . . + Pn dXn = px dxx +... +pn dxn,
and therefore by the reasoning of § 1 78,
Y Y P P
will be unconnected, and
aft = Xi, & = Pi} (i=l,...,n)
will be a homogeneous contact transformation.
That the functions Pl5 ...,Pn are known, when Xls ...,XW
are known, follows from the equations
2?<^=
2**^-
(k = 1, ...,n).
These equations could only then fail to determine P1,...iPn
uniquely in terms of xx, ..., xn, px,...,pn when all %-rowed
determinants of the matrix
axx
2>XX
axx axt
2>X
fy?
n
ax
n
~bX„ <>X„
ax
a&
n
tyl
vanish identically, that is, when Xv...,Xn are connected; and
as Xls ...,Xn are unconnected the equations do not fail.
x'30
PFAFFIAN SYSTEMS AND
[179
The problem then of finding a homogeneous contact trans-
formation is that of finding n unconnected functions of zero
degree in px, ...,pn, and mutually in involution; and to every
such system of functions one contact transformation scheme
will correspond. We have shown in Chapter XIV how this
problem depends on the solution of a complete system of linear
partial differential equations of the first order ; and we have
also seen how, when we are given m of the n functions in
involution, the remaining (n — m) are to be found.
Example. Any n unconnected functions of xx, ...,xn are in
involution and of zero degree ; the contact transformation
scheme, however, which corresponds to this solution of the
problem, will be a mere point transformation.
If on the other hand we take any ('ft— 1) unconnected
functions of plt ...,pn of zero degree they will be in involu-
tion ; as there cannot be more than (n— 1) such functions the
71th function of the involution system must involve xv ..., xn.
Let us take ±-I, ...,^ri as the (n—1) functions ; and let
Pn Pn „j
v be the nth function ; since it is in involution with — we
P*
7iX1
px <*V
Pn
must have — r *-k r — = 0 : we therefore have the fol-
Pl *xn
lowing equations to determine v :
^xx _ <>x2 _
and may take v to be the function
^n
Pn
p1x1+ ... ■{■ pn xn ^
Pn
We now have n unconnected functions in involution, and
of zero degree, viz.
y Pi Y — Pn-i y —PiXi~^m"~^~Pnxn
^1 — yy >'">^»-l— ~ ' ^n— -
Fn lJn t'n
The identity
Px dXx + . . . + Pn dXn = px dxx + ... +pn dxn
gives us
i =n — \ i = n i = n
KPn Pn
180] CONTACT TRANSFORMATIONS 231
and therefore
"n = Vnt ■* i = xiPn> •••> " n-1 = — xn-lPw
We thus have the homogeneous contact transformation
«/ _ ^1 V — -^"-1 V _ ??i #1 + • • » + ?JW #ra
Fn lJn lJn
Pi = ~ xiPn> •••> Pn-1 = ~xn-lPn> Pn ~ Pn'
§ 180. By a homogeneous contact transformation any
Pfaffian system is transformed into a Pfaffian system. For if
\1) Ji (%it ...5 X,n, pv "'iPii) — "> "•)Jnvcv •••> ^rj> ^is '••>Pn) = ^
are the equations of a Pfaffian system ; the contact trans-
formation
(2) x'{ = X{, & = P^ {i=\i...in)
will transform these equations into some other n equations, say
(«v H>i\xv •••! xni Pv •••>Pn) = 9 •••» Vnl^iJ •••>xn> ^ij ••'tPn) = "•
What we have therefore to prove is, that any consecutive
values of o^j •••»a4» JPij •••>i3»j satisfying the equations (3) will
satisfy the equation
p[ dx[ + ... +pfn dx'n = 0.
Now to two consecutive values of x[, ...,x'n, p[, •••,p/,l satis-
fying (3), there will correspond two consecutive values of
x1,...,xn, P\,...,pn satisfying (1); and therefore — from the
definition of a Pfaffian system — satisfying the equation
px dx± + ... +pn dxn = 0.
Since the transformation is a homogeneous contact one
p[ dx[ +... +p'n dx'n = px dxx+... +pn dxn = 0 ;
and therefore the equations (3) satisfy the definition of a
Pfaffian system.
If we know any integral of an assigned differential equation
of the first order, we know how to write down a Pfaffian
system which will include the assigned differential equa-
tion. If to this known Pfaffian system we apply any known
homogeneous contact transformation, the assigned differential
equation will be transformed into another equation, of which
we shall know the Pfaffian system, and therefore the integral.
It is at this point that we begin to see the advantage of
Lie's extended definition of an integral of a given equation.
232 DIFFERENTIAL EQUATIONS AND [180
The assigned differential equation may only have an ordinary
integral, that is, the Pfaffian system, which contains it, may
have only one generating equation ; yet possibly the equation
into which the differential equation is transformed will have,
as the Pfaffian system including it, one generated by two or
more equations.
It may even happen that by the contact transformation the
assigned differential equation is transformed into an equation
only containing x[, ..., x'n, that is, into a generating equation
of the Pfaffian system.
§ 181. Example. Consider the equation
2x2x3 px = xx p2p±
of which a complete integral is easily found, viz.
where a, 6, c are arbitrary constants.
Iif(x1, ..., xn) = 0 is an integral of an assigned differential
equation $ (#i, >•-, %n, Pi, •">Pn) — °> then this integral gives
us the Pfaffian system
/=o;
Pi _
¥~
£>2
Pn
-- df'
t)^!
tx2
clxn
and, since from the definition of an integral, </> = 0 is deducible
from these n equations, it must be one of the equations of
the system.
In the example before us it is then only necessary to add
two equations to the given differential equation and its
integral, in order to have a Pfaffian system ; the third equa-
tion which we could obtain would be connected with these
four.
We may take these equations to be
2ax3p1 + x1pi = 0
and 4 a2 x3p3 — (x22 + c + 2 bx3 + 4 a2xi) p± = 0,
and, by aid of the given integral, the second of these is thrown
into the more convenient form
4 a2 x32p3 — (bx3 + ax2) p4 = 0.
The Pfaffian system with which we are now concerned
is then
(1) ^2x32Pi2-xi2P2P4: = °>
(2) bx32 + cx3 + x22 x3 — ax2 + 4 a2x3 x4 = 0,
181] CONTACT TRANSFORMATIONS 233
(3) 2ax3p1 + x1p4 = 0,
(4) 4 a,2 x32p3 — (bx32 + axx2) p^— 0 .
If we apply to this system the contact transformation,
, _ Pi , _ p2 f _ P3 , V\ P-2 Pz
Pi Pi Pi Pi Pi " Pi
P'l=-Vl2h> P2=~X2Pi> P3=-®sPv Pl=Pl>
we obtain the four Pfaffian equations
(1) p'2p'ix2 + 2p'2p'2x'2 = 0,
r>'
(2) bp'2 - cp'3p\ -p'2 ^ - ap*
Pi
- *a2p3 (p[x[ +p2x'2 +p3x3 +p'iOQ = 0,
(3) 2ap3x'1+p'1= 0, (4) 4a2p32x3—bp32-ap[2 = 0.
Eliminating p[, p'2, p3, p\ from these equations, we obtain,
after a little labour, not one but two equations, viz.
4ft3^2-4a24 + 6 = 0, c-4ft4a/2+4ftV4 = 0.
It follows, therefore, that by the contact transformation we
pass from the equation
2 x2 x2p£ — x?p2 Pi = 0 5
with its ordinary complete integral
bx2 + cx3 + x22 x3 — axx2 + 4a2x3xi= 0,
to the equation Pi Pi %2 + %P%Pz xi — °>
with Lie's complete integral
4ft3#12— 4a2«3 + 6 = 0, c — 4ft4fl32 + 4ft2iC4 = 0.
Example. Any equation of the form
P1%1 + --.+PnXn=Pnf(^>'-->^r1)
Pn Pn
is transformed by the contact transformation
ry-Pi rJ -Pn-1 y _PiXi+ — +Pnxn
Jul—'Z~'"^'Ln-l— „ ' ■*» — n
Pn Pn Pn
Pi = ®iPn> •••' Pn-i = xn-iPn> Pn = Pn
into xn =f (aj15 ..., &n-i)-
This would not be a differential equation at all, according
234 THE TRANSFORMATION OF [181
to the usual definition, but is one in Lie's sense; and, since
we know a complete integral of it, viz.
X^ — CI j , . . . , XR — Cln ,
where a1: ..., an are constants connected by the law
an = j («15 ..., cin_l),
we at once deduce that
a1x1 + .. . + an_lxn_1 + xn =f(a1, ..., an_x)
is a complete integral of the given equation.
§182. The functions X1,...,Xn, P1,...,Pn which define
a homogeneous contact transformation satisfy the conditions
of beino- a complete homogeneous system of functions in
normal form ; for
(X15 PJ = (X2, P2) = ...= (Xn, Pn) = 1,
and all other alternants of the system vanish identically;
whilst Xx, ..., Xn are homogeneous of degree zero, and
Px, ..., Pn homogeneous of degree unity. _
If we are given two homogeneous function systems of like
structure %j ..., Ujn and vlt ..., vm,
we must now prove that they can be transformed, the one
into the other, by a homogeneous contact transformation.
If /j K, ..-, Um), ...,/w (uv •••> um)
are functions equivalent to ux, ..., um, and such that/15 ...,fm
are in normal form, we know that
/l \vl ' • • • ' *w ' • • • ' /m \vl ' • • • ' *W
will be a function system equivalent to v1 vm, and of
the same normal form as
Also if a contact transformation
4 = Xi? Pi = Pi, (i = h...,n)
transforms fj (vx, ..., vj into /,• (w15 ..., wj for all values of
the suffix j from 1 to m, that is, if
fj K>-.> O =/j K> ••" ttm)> 0' = *• -' m)'
where u'- denotes the same function of x[, ..., x'n) plt ..., pn
that i>- is of o?15 ..., 0Jn, £>la ..., pn, then will
vj = ui' 0' = 1> •••>*»)■
183] COMPLETE FUNCTION SYSTEMS 235
In order, therefore, to prove that two homogeneous function
systems of like structure are transformable into one another
by a homogeneous contact transformation, it will only be
necessary to prove that two such systems of the same normal
form are so transformable.
We have seen that to u1} ..., um we can add functions
um+1, ..., u2n, till u1? ...,u2n is a system of order 2n, con-
taining no Abelian functions, and in normal form ; these
2n functions will therefore define a homogeneous contact
transformation scheme. If we similarly add functions to the
system vx, ..., vm till it forms a complete homogeneous system
of order 2 n, containing no Abelian functions, and in normal
form, then vlt ..., v2n will also define a homogeneous contact
transformation scheme.
In these two systems Uj is homogeneous and of the same
degree in px, ...,2\ *na* vi 18> y*z- unity or zero; and when
we say that the two systems have like structure we mean
that u^ in one system corresponds to v{ in the other.
We may suppose that ux, ..., un are the functions of zero
degree, and un+x, ..., u2n the functions of degree unity ;
xi = ui> Pi = un+i> (* = 1, ..., w)
will then lead to
i = n i = n
and x'- = <•, p^ = v'n+i, (i = 1, ..., n)
i = n i = n
will lead to 2 Pidxi — 2 Pi dxi •
It follows that the equations
ui — ^> (* = 1"--5 2n)
i—n i=n
will lead to 2 Pi dxi = 2 Pi dxi 5
that is, the functions v1,...,v2n are transformable to the
functions uls ..., u2m by a homogeneous contact transforma-
tion scheme; and in particular v1,...,vm are transformable
into ult ..., um, Vj being transformed into u^.
§ 183, Having now proved that two complete homogeneous
systems of the same order and structure are transformable
into one another by a homogeneous contact transformation,
236 THE TRANSFORMATION OF [183
we shall now investigate the conditions under which it is
possible to transform any m given functions vlt ..., vm re-
spectively into the given functional forms uls ..., um, by
a homogeneous contact transformation.
Let x\ = Xt, p\ = Pi} (i = l,...,n)
be a homogeneous contact transformation ; we have
Suppose that this contact transformation transforms Vj into
Uj, where
Vj — (f>j {xx, ...,xn, px, ...,pn) and Uj = fj (x1} ...,xn, p>ii "">Vn),>
so that
jj (£C2 , .,., xn, Pi, ..., pn) = <f>j (Xi , ...,xn, p1} ..., pn) ;
then
J
that is, by the conditions for a homogeneous contact trans-
formation,
From the mere fact that U: = v'j we could not of course
conclude that u- = v'j ; we were only able to draw this con-
clusion from the forms of the functions Xr, ..., Xn, Plt ..., Pn
which define the homogeneous contact transformation.
Since u- = v'j, and Uj — v\,
Uj . Ui = v"j . v'f, and therefore (w-, Uj) = (Vj,V{) ;
and therefore the transformation, which transforms vt, ...,vm
into iix, ...,un respectively, must transform the alternant
(Vi,Vj) into the alternant (u^Uj).
183] ANY SYSTEM OF FUNCTIONS 237
Again since
to ~z to M +z to Wi ( " ' ""n)t
^ ** to ~^** to k- + * n to m '
and, as Xx, ..., Xn are of zero degree, and P15 ..., Pn of degree
unity, we therefore have
i=n i=n
The transformation then which transforms v$ into u^ must
also transform Pv$ into Pu^.
From these considerations we see that, given the functions
vi, ...,vm and uv ...,um, we must form the complete homo-
geneous systems of which they are respectively functions.
To do this form the alternants from vx, ...,vm and also the
functions Pv1, ..., Pvm ; if by this means we obtain no function
unconnected with vx, ...,vm the system is complete and homo-
geneous ; if, on the other hand, we obtain a new function
we add it to v1,...,vm, and proceed similarly with the new
system. As there cannot be more than 2n unconnected
functions of xx,...,xn, plt ...,p>n we mus^ thus ultimately
arrive at a complete homogeneous function system. When
we have formed the two complete homogeneous systems of
lowest orders which contain the given sets of functions, we
can tell whether or not the systems are of the same order and
structure; if they are, the given functions vx, ...,vm are
respectively transformable into u1,...,um by a homogeneous
contact transformation, but otherwise they are not so trans-
formable.
Thus any homogeneous function can be transformed into
any other of the same degree ; for the function group of each
is of order one, and the structure the same.
In particular, any homogeneous function u of degree unity
can be transformed into px ; and therefore the operator u can
be transformed into r — by a homogeneous contact trans-
formation if, and only if, u is of degree unity.
So if ux, ..., um are m unconnected homogeneous functions,
each of degree unity and mutually in involution, they can be
238 NON-HOMOGENEOUS [183
transformed into px, ...,pm, and therefore ux,...,um can be
transformed into- — 5 •••>;- — respectively.
§ 184. Although in considering the theory of Pfaffian
systems of equations it is much more convenient to work
with the homogeneous equation
p1dx1+ ... +pndxn = 0,
yet in particular examples, and in the cases n = 2, and n = 3,
it is often simpler to take the non-homogeneous form
(1) dz = pxdxx + ...+pndxn.
It is clear that to satisfy this equation we must have at
least (n+ 1) unconnected equations between
z, xx , . 1 . , xn , Pi , . . . , pn ,
but instead of considering this equation independently we
may deduce its theory from that of the corresponding homo-
geneous equation.
Jje<i z — Vn+v xi = 2/i» •••»*» = Vn>
(2) px = -^-,^pn=-
9n + l 9.n+l
where qn+x is not zero ; then the equation (1) is equivalent
to the homogeneous one
$!<%/!+... + qn+1dyn+1 = 0.
To satisfy this equation we must have (n + 1 ) unconnected
equations in yx, ..., yn+1, qx, ...5<7W+1 ; and in order that
■* 1 — &i j •••> ■* n+l — Oi,
n+l
may satisfy the equation, for all values of the arbitrary con-
stants, it is necessary and sufficient that Yx, ..., Yn+1 should
be (n+l) unconnected homogeneous functions of
in involution.
Let Z be the function in 2, xx, ...,xn,px, ...,pn equivalent
to Yn+X; and Xx, ...,Xn the functions which correspond to
Yx, ..., Yn respectively.
If jPis any function of yx, ...,2/n+u -^-» •••»-^2- > it is also
5w + l 9n+l
a function of a;ls ... ,xn,z, px, ...,pn, in which form we shall
184] PFAFFIAN EQUATIONS 239
denote it by $ : we then have F = <£, and
7>F d* *F 1 '4? 3$
7>F &* ^ 1 d* ,. , ,
-^ = -— . — = — , (i = l,...,n).
If we now denote the expression
i=n i=n
by [itj^a.^, we deduce that
(F;, Ffc) =--— [X<,Xft],
1 ^ A; = 1 fh' '
(Yn+i> Yk)y,q = — ~ [%> ^"J*
* * <Zn+l
We conclude therefore that the necessary and sufficient
conditions, in order that
Z = &m+i> Xj = ax, ..., Xn = an
may satisfy the equation
dz = p1dx1 + ... +pndxn,
[z, x{] = o, [x{, xk] = 0, dz1^ ;;;; ™) •
If two functions u and v of the variables z,x1,...,xn,
Pi>"->Pn are suc^ that [^v]z,z,v — °> we say they are in
involution. Similarly we say ' that two equations u = 0,
•y = 0 are in involution if the equation [u, v]^ 8 _ = 0 is
connected with u = 0, v = 0. We generally omit the suffixes,
and write [u, i>] for [u, v]2)2; „, the variables z, xlt ...,xn,
Pi,...,pn being understood.
The equations Z = 0, Xx = 0, ...,X„ = 0,
will then, and only then, satisfy the Pfaffian equation
dz — pxdxx + ... +pndxn)
when they are unconnected and in involution.
It follows that (n+ 1) unconnected equations in involution
cannot all be equations in x1,...,xn, Pi,.*.,pn only, but must
contain z ; else would they not lead to
dz = p1dx1 + ... +pndxn.
240 NON-HOMOGENEOUS [184
We may prove this last result independently thus ; suppose
Z =0, Xx = 0, ...,Xn = 0
do not contain z, we then see that
[Z,X{] = (Z,X{) and [X„Z,] = (X-, Xlc) ;
we now have the (n+ 1) unconnected differential equations
(Z,f) = 0, (X1,f) = 0,...,(Xn,f) = 0,
with the (n+ 1) unconnected integrals
Xx — ax = 0, ..., Xn — an = 0,Z—an+1 = 0,
and this is impossible, the equations being in 2n variables
only.
§ 185. Suppose we have (n+ 1) unconnected functions of
z, xx, ..., xn, px, ...,pn in involution, viz. Z, Xx,..., Xn. If we
apply the transformation (2) of § 184, the identities
[Z,X{] = 0, [XitXk] = o
are transformed to
We have therefore (%+ l) unconnected functions of
2/i» •••» Vn+i> 9i> "•> 9.n+i>
homogeneous and of zero degree in qx, ..., qn+1, and in in-
volution. We can therefore write down the homogeneous
contact transformation
V'i = ?i> 4i = Qi, (*=1,. ..,%+!);
i = n+l i = n+l
and, since 2 si <% = 2 ft <fyi »
we see that, if
Pi = -jft-, (i=l,...,n),
dl/n+i-pi dijx-... -Pn dy'n = ^±1 (cfe-^ (fa?!— ... -p» *»*)•
Therefore
(i) • = zI'fl4 = x<fj/< = p4 (* = i, ...,»)
will be a transformation, with the property
cfe' — £>i J^ — , . . — p'n dx'n = p (dz —px dxx — . . . — pn dxn),
186] CONTACT TRANSFORMATION 241
Q
where p = p^+- > and is therefore a homogeneous function of
2/n •'•> 2/ra+i> Qn •'•> °n+i> °f zerP degree, and therefore a func-
tion of z, xlf..., xn, px,...,pn.
A transformation such as (1) is called a contact trans-
formation; and we see that, when we are given the (n+1)
unconnected functions in involution, viz. Z, X1,...iXn, the
contact transformation is entirely given.
The functions Px, ..., Pn, as well as the factor p, may be
obtained algebraically from the equations
IZ %\*Xi_
^-2^ = 0. (Jc=l,...,n).
The contact transformation
z'=Z, x\ = X{, p\=Pi, (i =!,...,»)
has the property of leaving the Pfaffian equation
dz—p1dx1—...—p>ndxn = 0
unaltered ; and therefore — from the general definition of a
group — the system of all contact transformations, regarded as
transformation schemes in the variables z,x1} ...,xn, px, ...,pn,
generates a continuous group, though of course not a finite
continuous group.
§ 186. Example. The variables being y1, y2, y3, qx, q2, q3,
and uv u2, v1} v2 being unconnected homogeneous functions of
zero degree, such that every function of u15 u2 is in involution
with every function of vx,v2i but ux not in involution with
u2, nor v-l with v2, it is required to find simple forms to which
these functions may be reduced by a contact transformation.
The alternant (ux, u2) is of degree minus unity, and cannot
therefore be a function of ux and u2 ; we have therefore three
unconnected functions ux, u2, and" (ult u2) ; and, as vx is in
involution with ux and u2, it is also in involution with
(u1, u2). We thus see that ux, u2 and (ux, u2) are three
unconnected functions of a homogeneous system ; and that
there are at least three unconnected functions in involution
CAMPBELL JJ
242
APPLICATION OF THEORY
[186
with each of these functions, viz. vlf v2 and (vx, v2) ; it there-
fore follows (since the number of variables is six) that there
cannot be more than three functions in the system containing
ut, u2 and (ux, u2). The conclusion we draw is that ux, u2,
(u-^u.) form a complete homogeneous function system, and
that i\, v2, (i\, v2) is its polar system.
Since ux,u2, (u1, u2) is a system of order three, it must
have at least one Abelian function. We see this by recalling
the normal form of a complete system ; or we may prove it
independently by writing down the contracted operators of
a complete system of order three, when, since the Pfaffian
determinant' 0 ? (Ul, u.2), (uls u3)
(u2, uj, 0 , (u2, u3)
(u3, ttj), (u3,u,), 0
vanishes identically, we see that not more than two of the
contracted operators can be unconnected.
If all two-rowed minors of the above determinant vanished,
then all the functions would be in involution ; there must
therefore be either three or only one Abelian function.
In this example, since (u15 u2) is not zero, there must be
one, and only one, Abelian function ; and, as it is not a mere
function of ux and u2 (for then ux and u2 would be in
involution), it is not of zero degree (see § 165). When the
system is then reduced to normal form it is of like structure
with %,&.?.;
and can therefore be reduced to this form by a homogeneous
contact transformation.
We can therefore, by a homogeneous contact transformation,
so reduce ux and u2 that each will be a homogeneous function
of yx, <7j , q3 of zero degree.
Since i\ and v2 are homogeneous functions of zero degree,
in involution with every function of ux and u2, they are in
involution with yx and — . Since they are in involution with
ylt they cannot involve q1 ; and, since they are also in involu-
tion with — , we see that they cannot involve yx or yy We
conclude therefore that vx and v2 are homogeneous functions
of y2 , q2, qz of zero degree.
If we now take
H3 ^3
187] TO AMPERE'S EQUATION 243
we see that ux and u.2 can be transformed by a contact trans-
formation so as to be functions of xx and px ; while by the
same contact transformations vx and v2 become functions of
x2 and p2
§ 187. The above example has an important application to
Ampere's equation,
Rr + Ss + Tt+U(rt-s2) =V.
If this equation admits the two systems of intermediary
(where / and cp are arbitrary functional symbols), then we
know (Forsyth, Differential Equations, § 237) that
Oj, v2] = 0, [u1? vj = 0, [u2, vx] = 0, [u2, v2] = 0.
From what we have proved, we see that, when we have
applied a suitable contact transformation to the original
variables, we may take ux and u2 to be functions of x and p
only. Now by a contact transformation any equation of
Ampere's form is transformed into some other equation of the
same form. In the new variables then, Ampere's equation
has an intermediary integral
Hi =f(uz)>
where ux and u2 do not involve y, z, or q.
This equation is therefore to be the result of eliminating
the arbitrary function from
dw, OU, ... . ,011^ 0U9s
^~ + r -^ =f(u9) (~+r — -2) ,
ox op v ox op J
^Ul Ml \ ^U2
The eliminant is
iouxou2 ouxou2\
^ ox op op ox' '
and, as ux is not a function of u2 , we cannot have
oux ou2 dUj ou2
ox op op ox '
so that the equation must be s = 0. This is therefore the
B 2
244
SPECIAL CASE
[187
form to which an equation of Ampere, admitting the two
systems of intermediary integrals
^i-/K) = ° and Vi— <i> (»J = °>
can be reduced by a contact transformation.
An interesting proof of this theorem of Lie's is given in
Goursat, Equations aux derivees partielles du second ordre,
I. p. 39.
If in the equation Rr + Ss + Tt + U (rt—s2) = Fwe have
S2 = 4 (RT + UV), there can only be one system of inter-
mediary integrals, u1=f(u.J). We now have, however,
[ii1} %2] = 0 ; for, since the roots are equal in the equation
A2 (RT + UV) -\US+U2=0,
we have (Forsyth, Differential Equations, § 238) ut = v.2 ; and,
since \ux, v2] = 0, we must therefore have in the limiting
case \ux, u.2\ = 0.
We now take ux = p, u2 = q ; and we see that p —f(q)
can only be an intermediary integral for all forms of the
function if the equation is
(rt-s2) = 0.
This then is the form to which this class of Amperian
equation, with the intermediary integral ux—f(u2) = 0, can
be reduced by a contact transformation.
CHAPTER XVII
THE GEOMETRY OF CONTACT TRANSFORMATIONS
§ 188. If the equations denning a contact transformation are
(1) 2! = Z, x't = Xi: p'i = P^ (i = 1, ..., n),
we know that the (n+1) functions X15 ..., Xn, Z form a
system in involution ; and conversely, when we are given
any involution system, we know how to construct a contact
transformation scheme.
In this chapter we shall show how contact transformation
schemes may be constructed without previously constructing
involution systems.
If we eliminate px, ..., pn from the (n+1) equations (1), we
shall obtain at least one equation of the form
/ (*^l» •••J *^7l' ^> "&L> *••' *k»J ^ ) = " I
and we may obtain 1, 2, ..., (n+1) such equations. We call
these equations the generating equations of the contact trans-
formation scheme.
Suppose that we have s generating equations, viz.
A =o, ...,/, = o,
then the equation
i =n i=n
(2) dsf - 2 Pi dxi ~ P (dz ~ 2 Pi dxi) = °
must be of the form
(3) *<&+•■; +P.<%=°i
where p1, ..., ps are undetermined functions of the coordinates
of corresponding elements.
We have, by equating the coefficients of dx'i}
-tf = ft §+••• +/>.§■
Similarly we obtain other identities by equating the coeffi-
246 THE GENERATING EQUATIONS OF [188
cients of dz'.dz, and so on; and we thus have (2n + 2 — s)
equations between the coordinates of corresponding elements
when we eliminate the undetermined functions.
If we add to these the s generating equations and eliminate
p, we shall have (2n + 1) equations connecting the coordinates
of corresponding elements.
These (2^+1) equations must be equivalent to the
system (1). For they are deduced from (1) and the Pfaffian
equation (2), which itself follows from (1) ; they are also
unconnected, since they satisfy (2) ; finally therefore, being
(2n+ 1) in number, unconnected, and following from (1), they
are equivalent to (1).
The generating equations alone can therefore determine the
contact transformation scheme ; and it is from this point of
view that we shall study them in this chapter.
§ 189. Any s equations connecting the two sets of variables
may in general be taken as generating equations. They must
however satisfy two conditions, viz. firstly the s equations,
together with the (2n+ 1 —s) derived equations, must be such
that we can by means of them express %[, ..., x'n,z', p'ls ...,p'n
in terms of xx, ..., xn, z, px, ..., pn ; and secondly we must be
able to express xx, ..., xn,z, px, ...,pn in terms of
£Cj , . . , , Xn , Z , Pi , , i , j jJn .
These two conditions are however equivalent ; for suppose
that from the assumed system we deduce
(1) z' = Z, x\ = X{, 2>i = Pi, (i = 1, ..., n),
then by the method of formation of the system we must have
dZ—^Pi dXi = pidz — ^Pi dxj).
Now p cannot be zero: for if it were the equation (2) of
§ 188 could not lead to (1), but must lead to exactly
(n+1— s) equations connecting x[ , . . . , x'n , z\ p'x , . . . , p'n . Since
then p is not zero, the functions Z, Xlt ...,X1V Px, ..., Pn must
(by § 178) be unconnected ; and therefore
xx , . . . , xn , z, px , . . . , pn
can be expressed in terms of x[, ..., x'n, z\ p'x,...,p'n.
191] A CONTACT TRANSFORMATION 247
§ 190. If we are given s equations which cannot be used as
generating equations of a contact transformation scheme, what
special property will distinguish these equations 1 We shall
call such a system of equations special equations. From
8 special equations we can, as in the general case, deduce
(2n+\—s) other equations; and these equations will be un-
connected, and will satisfy the Pfaffian equation
i = n i = n
dz' — 2 Pi dx'i = P (dz — 2 Pi dxi)-
If in the special equations we keep x[,..., x'n,zr all fixed,
that is, if we regard this set of variables as parameters, the
special equations together with the derived equations will
form a system satisfying Pfaff's equation
i = n
(1) dz—^drpidxi = 0.
If we now consider how the (2n+l— s) derived equations
were obtained, we shall see that we can eliminate^, ...,p'n,
and obtain exactly (n+ 1—s) derived equations not involving
these quantities ; these taken with the s special equations will
satisfy Pfaff's equation (1).
From that property of the equations, which makes them
incapable of being taken as generating equations, we see that
we must be able to eliminate the coordinates
£Cj , . . . , Xu , Z , rp\ ) • • • 5 Pn '
and so obtain at least one equation connecting
x-^ , . . . , xn , z, Pi , . . . , pn .
Suppose we thus obtain r equations
(2) fa (a^,. ,.,xn) z, p1,...,p1) = 0, (S=l, ..., r);
then for all values of the parameters x[, ...,x'n,z' the equations
will be the generating equations (and therefore, in Lie's sense,
an integral) of an Mn satisfying the system of differential
equations (2) (see § 155).
§ 191. We shall now limit ourselves to the case of n = 3
which offers the most interesting geometrical applications of
contact transformation theory.
248 CONTACT TRANSFORMATIONS WITH [191
We take x, y, z as the coordinates of a point, and x, y, z, p, q,
as the coordinates of an element in one space ; and we take
x', yr, z', p', q', to be the coordinates of the corresponding
element in the other space.
There may now be 1, 2, or 3 generating equations.
We first take the case where there is only one generating
equation.
Let this equation be
4> (x, y, z, x, y', z') = 0.
We now know that the Pfaffian equation
dz? —p' 'dx' — q' dy' ' — p (dz—p dx — q dy) = 0
is of the form dcp = 0 ; and therefore we get as the equations
defining the contact transformation scheme
0)
dch deb
p -r + * = o,
dz dx
deb deb
q x* + ~ = 0,
^ dz dy
dcp dcp
dcp
*'(i? + S) = 0' «'(5? + S7) = 0' *-*
dcp
7
The condition, that the coordinates of one element can by
aid of these equations be expressed in terms of the corre-
sponding element, shows that the three equations
»*+(3 + 3>-*
/ deb 3(/)\
must be unconnected in the variables x' ', y', zf.
It follows therefore, after some simple algebraic reduction,
that the determinantal equation
d2cp
da? da/
d2cp
d2cp
bep
dx
dx ~by'
dx dz'
d2<p
d24>
d2ep
i(j)
(3)
dy dx'
d2cp
dydy''
dy dz'
d2cp
dy
dcp
= 0
dz dx'
dzby'
dz dz' '
Tz
dcp
dcp
dcp
r\
M *
w ■
dz~' '
0
must be unconnected with 0 = 0.
We could not therefo]
re take as a generating equation
xx' + yy + zz' = 0,
192] ONE GENERATING EQUATION 249
for the determinantal equation, formed from it, would be
connected with it, as may be easily verified.
This is an example of a special equation ; the M2 defined by
the equations
xx' + yy' + zz' — 0, pz' + x' = 0, qz* + y' = 0
must therefore be such that we can eliminate x, y', z' from
these equations ; if we do so, we obtain the equation
px + qy—z = 0,
which is satisfied by
xx' + yy' + zz" = 0,
for all values of the parameters x' ', y', z* .
From the symmetry of the equation (3) in the two sets of
variables x, y, z and x' ', y', zr, we verify the theorem of § 189
as to the equivalence of the two limiting conditions, imposed
on the general arbitrariness of the generating equations.
§ 192. If <p = 0 is a generating equation of a contact trans-
formation scheme, the determinantal equation (3) of article
§ 191 will be unconnected with 0=0. If then we eliminate
x\ y', z' between the equations (2) and (3), we shall obtain an
equation connecting x, y, z, p, q. Elements satisfying this
equation will be called special elements.
The equations (1) of § 191 will in general determine one
definite element x\ y\ zr, p', q' to correspond to each element
x, y, z, p, q. If, however, x, y, z, p, q are the coordinates of
a special element it will not have a definite element corre-
sponding to it, but an infinity of elements. Similarly, we
shall have special elements in space x', y', z' .
A particular system of special elements may be obtained
thus : eliminate x', y\ z' from the equations
cxb cSd) <)(h
the resulting equation in x, y, z is known as the special
envelope of
4> (x, y, z, x', y\ d) = 0,
x\ yf, z' being regarded as parameters.
The element consisting of a point on the special envelope
together with the tangent plane at the point will be a special
element ; to this special element there will correspond an co2
250 THREE DIFFERENT CLASSES [192
of elements, consisting of the point x', y', z1 together with the
ocr of planes through this point.
§ 193. There are three different classes of element manifolds
in three-dimensional space. There is, firstly, the manifold M2
generated from one equation only ; such a manifold we shall
call a surface M2.
Let fix, y, z) = 0, p J- + ^ - 0, q / + ~- = 0
J v ' a ' 1 hz <>x l Iz Zy
be the Pfaffian system of a surface M2 ; and let
4> (x, y, z, x', y\ z') = 0
be the equation which generates the contact transformation
scheme.
The generating equation (or it may be equations) of the 31. 2
which corresponds in the space x', y', z' is that one obtained
by eliminating x, y, z from the four equations
7)X ~bz ~2)X ' ~bz ty dz ~ <)y <iz '
If we regard x, y, z as variable parameters connected by the
equation f(x, y, z) = 0, the generating equation is therefore
the envelope of
<f> (x, y, z, x', y', z) = 0.
The manifold M.2 with two generating equations we call a
curve M2.
Let the Pfaffian system of a curve M2 be
/i (x> y> z) = °> f% (x> y> z) = o,
and the equation obtained by eliminating A : ^ from the
equations
L v hz lZJ <>X <iX
that is, the Pfaffian system
The generating equation of the i/2, which corresponds in
194] OF ELEMENT MANIFOLDS 251
space x\ y\ z', is therefore obtained by eliminating x, y, z from
a=o, a = o, 3#44)=°-
D(x,y,z)
This generating equation will be the envelope of
<})(x,y,z,x',y', z'),
where the parameters x, y, z are connected by
/i (x, y, z) = 0, f2 (x, y, z) = 0.
The manifold M2, which consists of the fixed point a, b, c
with the co2 of planes through it, has as the generating equa-
tion of the corresponding M2 in space x\ y', z' the surface
§ (a, b, c, x\ y\ z') - 0.
§ 194. If two surface manifolds have a common element
they must touch ; if two curve manifolds have a common
element they intersect ; and if a curve manifold has an element
common with a surface M2 they also touch.
If a point M2 has an element common with a surface M2
or a curve M2, the point must lie on that surface, or on that
curve ; but two point manifolds cannot have any common
element, unless they coincide entirely.
If then in space x, y, z two different iH2's have a common
element, the il/2's in space x', y\ z' which correspond to these
will also have in general a common element ; the exceptional
case is when the first common element is a special one.
Thus, if the two surfaces
<f> (x, y, z, ax, bx, cj = 0 and $ (x, y, z, a2, b2, c2) = 0
touch, the common element must be a special one for the
contact transformation with the generating equation
$ (x, y, z, x\ /, zf) = 0.
For otherwise the M2 consisting of the point av blt ct with
the oo2 of planes through this point would have a definite
element common with the point M2 whose coordinates are
a2, b2, c2, and this is of course impossible.
So if two ilf2's have an infinity of common elements, the
corresponding surfaces will also generally have an infinity of
common elements.
Thus, if two surface il/2's have an infinity of common
elements, they must either touch along a common curve ; or
have a common conical point, and the same tangent cone at
25.2
RECIPROCATION A CASE
[194
the conical point ; if the corresponding M2's in the other space
are also surface manifolds they must also have one of these
properties.
Again, if a curve A is traced on a surface B, then if A is
transformed to a curve A', and B to a surface B\ we must
have A' traced on B' ; if, however, A is transformed into a
surface A' and B into a surface B', the two surfaces A' and B
must either have a common conical point, with a common
tangent cone at it, or they must touch along a common curve.
Again, if A and B are two points, then the straight line
joining these points will be a curve M2, with one infinity of
elements common to the point manifold A, and another infinity
of elements common to the point manifold B ; if then this
straight line M2 is transformed to a curve M2 it will be the
curve common to the two surfaces A' and B' ; if, however, it is
transformed into a surface M2, it will generally be a surface
touching A' along one curve, and B' along another curve.
§ 195. The most interesting example of contact transforma-
tion of the first class is obtained by taking the generating
equation 0 = 0 to be linear both in x', y', z' and in x, y\ z, viz.
x (at x' + b1y' + ct z' + d1)+y (a2 x' + b2y' + c2 z' + d2)
+ z (a3 x' + b3y' + c3 z' + <£,) + a4 x' + b±y' + c4 z' + d± = 0.
We see at once that the only limitation placed on the con-
stants in this equation, in order that 0=0 may generate a
contact transformation, is that the determinant
a
a,
2>
'3'
'4>
33'
?4»
d,
d2
d3
dA
should not vanish.
If this condition is satisfied the equation $ = 0 will generate
a contact transformation ; and, since the determinant does not
contain any variables, the contact transformation will be one
with respect to which there are no special elements.
Clearly a point in either space will correspond to a plane in
the other ; and the straight line given by
a1x + (31y + y1z + S1= 0, a2 X + fi.,y + y2 Z + b.2 = 0
will be transformed to an M2 whose generating equation is
the envelope of the plane <f) = 0, when we regard x, y, z as the
parameters. This envelope is a straight line, and therefore
196] OF CONTACT TRANSFORMATION 253
the contact transformation transforms straight lines into
straight lines.
If we take as the generating equation
0 = xx' + yy' ' —z—z' — 0
— a form to which any equation, linear both in x, y, z and
x',y\ z\ can be reduced by a projective point transformation —
we have the well-known contact transformation
p = x, q = y', p' = x, q' = y, sf = px + qy—z]
this is geometrically equivalent to reciprocation with respect
to the paraboloid of revolution
2z = x2 + y2.
§ 196. We now proceed to discuss at greater length the
second kind of contact transformations, viz. those in which
there are two generating equations.
Let these equations be
<j> (x, y, z, x, y', z') = 0, \fr (x, y, z, x', y', z') = 0 ;
then, since the equation
dzf—p'dx' — qdy'—p{dz—pdx — qdy) = 0
is to be of the form
kd<b + y.d^ = 0,
we must have
f)z }>z' hy dy
If we eliminate the undetermined function A. : \x from these
equations we shall have three equations ; and these, together
with the generating equations, determine the contact trans-
formation scheme.
The equations (b = 0, yfr = 0, may be any whatever, provided
that the above five equations determine an element of one
space in terms of the corresponding element in the other
space.
254 CONTACT TRANSFORMATIONS WITH [196
If we take W to denote A0 + ^0, and in differentiating
regard X and ju- as mere constants, we may express this
limitation by saying that the four equations
0 = 0, 0 = 0, p
5F ZW
5z
+
*w *w
Zx
oz <iy
in the variables x' ', y', z', A. : /u. are unconnected.
It may be proved without much labour that this condition
is equivalent to saying that the determinant
30
^0
i<i>
ix
Mj
i>z '
50
50
50
2>x
ly
Iz '
WW
WW
WW
ttx^x'
^>y~&x
t>Zi)x'3
WW
~dxc*y'
ww
WW
Zyty'J
Tizlsy''
WW
<*X~dz'
WW
'by'bz
WW
Tszlsz'*
0 >
50
^0
50
5?
0
50
30
W
50
5?
must not vanish by aid of 0 = 0, 0 = 0 for all values of
A : fx ; that is, the determinantal equation must be unconnected
with 0 = 0, 0- = 0.
If we substitute in this determinant for x\ y', z* , A. : ju, their
values in terms of x, y, z, p, q obtained from (1), and equate
the result to zero, we shall have the equation satisfied by
special elements in the space x, y, z.
§ 197. In accordance with § 190, we notice that the limita-
tion placed on the generating equations is that 0 = 0, 0 = 0
must not be, for all values of the parameters xf, y', zf, the
integral of any partial differential equation of the first order.
Example. It may at once be verified that we could not take
as generating equations
axx' + byy' + czz' = 0, xx' + yyr + zz* — 0.
If, regarding x\ y' , d as parameters, we complete the Pfaffian
system of which these are the two generating equations, we
have as the third equation
(a — b) x'y' — (b—c) py'z! +(c—a) qx'z'.
198] TWO GENERATING EQUATIONS 255
Eliminating the parameters x', y', z' we get as one of the
equations of the Pfaffian system
z = px + qy;
and we see that, according to Lie's definition,
axx' + byy' + czz' = 0, xx' + yy' + zzf = 0,
is therefore a complete integral of
z = px + qy.
In this, as in all classes of contact transformations, the
general principle holds that two ilf2's with a common element
are transformed into two M2'a with a common element.
§ 198. Before proceeding to discuss the applications of this
class of contact transformations to geometry, we write down
some elementary properties of complexes of lines, which will
prove useful in the sequel.
We take as the coordinates of a line whose direction cosines
are I, m, n, and which passes through the point x', y\ z'
I, m, n, a, f3, y,
where
a = mz' — ny\ fi = 7ix'—lz', y=ly' — mx'.
If the coordinates of a line are connected by the linear
relation a I + fi'm + y'n + Va + m'/3 + n'y = 0,
where l\ mf, n', a, /3', y are any given constants, the line is
said to belong to a given linear complex ; V, mf, n', a, (3\ y
are said to be the coordinates of the complex. If the coor-
dinates of the complex are connected by the equation
I'a' + m'p' + n'y' = 0,
then the coordinates of the complex are the coordinates of
a line, and the complex consists of straight lines intersecting
a given line.
WTe may take l\ m' , n' to be forces along the axes of
coordinates ; and a, ft', y to be couples whose axes coincide
with the axes of coordinates. If a rigid body is rotated about
the line I, m, n, a, (3, y through a small angle dt, it has linear
displacements adt, fidt, ydt along the axes, and rotations
Idt, mdt, ndt about them. The work done by the given forces
and couples is then
(Va + m'/3 + n'y + la+m(B' + ny) dt ;
256 LINEAR COMPLEXES [198
and therefore, if a body is rotated about any line of the
complex, the given system of forces do no work on it.
These statical considerations enable us to simplify the
equation of a linear complex ; for, if we take the wrench
equivalent to the given system of forces and couples, we know
that it acts along a fixed line, which we now call the axis of
the complex ; let k be the ratio of the couple to the force
in the wrench, and let us take the axis of the wrench as the
axis of z. We now have
l'= 0, m'=0, a = 0, /3' = 0, y — kn\
and therefore, if a line be such that the wrench does no work
on a rigid body rotating about it, its coordinates must satisfy
the equation
y + kn = 0 ;
this therefore is a form to which any given linear complex
can be reduced.
An infinity of lines can be drawn through any point
x, y, z' which shall belong to the complex y + kn = 0 ; these
lines all lie on the plane yx'—xy' + k (z' — z) = 0, which is
called a null plane of the complex. Through every point
a null plane can be drawn.
Any two lines, whose coordinates are
I, m, n, a, /3, y,
I, m, -~, a, /3, — kn,
are said to be conjugate to one another with respect to the
complex.
If x', y\ rf lies on any straight line the conjugate line lies
on the null plane of x', y', z' '; and the null planes of two
points intersect on the line conjugate to the join of the two
points.
If the coordinates of two complexes
199] BILINEAR EQUATIONS 257
are connected by the equation
ha2 + hai + mA + m2/3i + niY2 + Wi = °>
they are said to be in involution. The statical interpretation
is that a wrench along the axis of one complex does no work
in a rigid body, which is moved along the screw of the other.
The two comp]exes, whose coordinates are respectively
V, m\ n', a', (3', /,
V, m', ^-, a, ft', —kn',
are said to be conjugate with respect to the complex
y + kn — 0.
If a line belongs to any complex, its conjugate line belongs
to the conjugate complex.
If two lines intersect, their conjugate lines also intersect.
A line coincides with its conjugate, if, and only if, it belongs
to the complex, with respect to which the lines are conjugate.
§ 199. Let us now take as our generating equations for the
contact transformation the bilinear equations
x {axxf + bxy + cxz' + d1) + y (a2x' + b2y' +...)
+ z(a3x' + b3y/+...) + a4x' + b^y/ + ... = 0,
x(a1x' + (31y' + y1z' + h1) + y(a2x' + p2y'+...)
+ z(azxf + p2,tf + ...) + a4:xf+ P4ty'+ ... = 0.
If we keep x', y\ z' fixed, these are the equations of two
planes; in order to simplify the form of the equations by
a projective transformation, we consider the positions of the
point x',y' \z', which will cause these planes to be coincident.
For the coincidence of the planes we must have
T • • • C^4 X T . • .
. v a^x' +bxy' + 0^' + dx a2x'+... azx
* a1x/ + j31y, + y1z/ + b1"a2x/+..."a3x'-
a^x +
equating these equal fractions to A, and eliminating x\ yf, z\
we have
ax — Aa15 b1 — \j31, c1 — \y1, d1 — Xbx
62 — A/32, c2 — Ay2, d2 — kb2
d3 — A83
a2-
-Aa2,
a3-
-Aa3,
<V
-Aa4,
3'
63-A/33, c3-Ay.
64-A/34, c4-Ay4, d4-A54
= 0.
CAMPBELL
258 SIMPLIFICATION OF THE [199
There are in general, therefore, four positions of the point
x', y', z', for which the generating equations will represent
the same plane.
We first consider the case where the four points lie on the
same plane ; and, by a projective transformation, we may
take this plane to be the plane at infinity.
The points therefore which give coincident planes must
satisfy the equations (1), when in these we put
dt = 0, 8j = 0, d2 = 0, b2 = 0, ... ;
and therefore all three-rowed determinants must vanish in the
matrix
ttx — \ax ,
ci2 — Aa2 ,
a3~^a3>
<z4— Aa4
&1-A&,
&2-*Ai>
h-W3>
&4-A/34
Ci-Ay19
c2-ky2,
c3-ty3>
c4-Ay4
Now these are cubic equations in A, and by hypothesis they
are satisfied for four values of A ; they must therefore be
identically true for all values of A.
The deduction of the necessary relations between the con-
stants, involved in these identities, is made easy by a geo-
metrical representation.
We take Ax to be a point whose coordinates are av bx, cx,
Bx to be the point whose coordinates are als /319 y1, and so on.
Taking A = 0 we see that Ax, A2, A3 are three collinear
points; taking A to be infinite we see that Bx, B2, B3 are
collinear. It now follows, from the given identities, that any
three points which divide the three lines Ax Bx, A2 B2, A3 B3,
in the same ratio are themselves collinear. These three lines
must therefore be generators of a paraboloid of which two
generators (of the opposite system) are Ax A2 A3 and Bx B2 B3.
It follows that A3 divides Ax A2 in the same ratio that B3
divides Bx B2.
Similarly we see that Ax, A2, A3, A± are four collinear
points dividing their line in the same ratios that Bv B2,B3, B±
divide their line.
§ 200. If we now take
X'= axx' + bxy' + cxz\ F'= a2x' + b2y' + c2z',
Z'= a1af + piy' + :y12f, W= a2x' + ^y'+y.z',
we see that the generating equations must be of the form
x(X' + d1) + y(T + d2)+z(pX' + qY' + d3)+p'X' + q'Y' + ^ = 0,
x(Z' + bj)+y(W' + bJ+z(pZ' + qW' + bJ+p'Z' + q'W' + b4=0,
where p, q, p\ q' are some constants.
201] BILINEAR EQUATIONS 259
We further simplify these equations by taking
y _ x+pz + p' „ _ dxx + d2y+d3z + c?4
" y+gr + q" ' y+qz+q'
Wr=b1x + 82y + b3z + bi
y + qz + q' "'
when we have as generating equations
XX'+Y' + Z=0, XZ'+W'+W=0,
where Xf, Y, Z\ W are connected by an identity of the form
aX'+bY'+cZ'+dW'-Q.
If finally we take new sets of variables x, y, z and x', y\ z',
given by
x = X, z=aZ+cW, y=-bZ-dW,
x' + iy'=aY' + cW, x' -iy'=bX' + dZ', z' = bY' + dW,
where i is the symbol v— -1, the generating equations reduce to
xz' + z + x' + iy' = 0, x(x'—iy') — z/ — y = 0.
To sum up : when the four points in space x\ y\ z' which
make the generating equations coincident are coplanar, the
generating equations can by a projective transformation be
thrown into the standard form
xz' + z + x' + iyf = 0, x (x'—iy) — z'—y = 0.
In this standard form we now see that every point has
this property which lies on the intersection of the cone
with the plane at infinity ; that is, any point on the absolute
circle at infinity has the property of making the generating
equations coincident.
§ 201. We must now study the contact transformation with
these generating equations
(1) x' + iy' + xz' + z — 0, x(x'—iy') — y—zf — 0.
It is to be noticed that, as the equations are not symmetrical
in the coordinates of the two spaces, the relation between the
corresponding elements in the spaces will not be symmetrical.
s %
260 AN IMPORTANT CONTACT [201
In addition to (1) we have for determining the trans-
formation
p' (x— q) + l +qx = 0, q' (x — q) +i {\ —qx) = 0,
p + z' + q{x'—iy') = 0;
and we see that each element in space x\ y', z' can be uniquely-
determined in terms of the corresponding element in space
x, y, z.
If, however, we wish to express x, y, z, p, q in terms of
x', y\ z', p\ q', we have, to determine x and q, the equations
p' + iq' 2 .
qx=—, — r^' q — x =
p—iq- ■* p —iq'
and therefore two different elements in space x, y, z will have
the same correspondent in space x' ', y\ z.
Such a pair of elements in space x, y, z we shall call con-
jugate elements ; it may easily be proved that the contact
transformation
x'=-q> y'=p, p'=y, q'= -%, z'= z—px-qy
will transform any element to its conjugate element.
Example. Prove that this contact transformation is the
result of first reciprocating with respect to xy = 2 z, and then
reflecting the surface with respect to the axis of y.
Reciprocation is equivalent to taking as our generating
equation xy' + yx' —z — z'= 0 ;
and therefore
«'=?> y'=P> z'= px+py-z, p'=y, q'=x.
If we now reflect with respect to the axis of y, we have
3"=-*', x"=-x\ y"=y'-,
and completing the contact transformation, generated by these
three equations, we have
p"=p', q"=-q'>
so that
z"=z-px-qy, x"- -q, y"=p, p"=y, q"= -x.
Example. Prove that if the element x, y, z, p, q is rotated
90° round the axis of z, in the positive direction, and the
conjugate element x', y\ z',p\ q' is reflected in the plane 0=0,
202] TRANSFORMATION 261
the two resulting elements will be reciprocal with respect
to x2 + y2 = 2 z, that is, will be connected by the equations
z + z'=px + qy, x'=p, y'=q, x = p\ y = q'.
§ 202. To the point x', y', z' there will correspond in space
x, y, z the straight line given by the generating equations
when we regard x\ y', z' as fixed. The only exceptional case
is when x\ y' ', z' lies in the absolute circle in its space, and
then we have as its correspondent a plane in the other space.
The six coordinates of the straight line corresponding to
x\ y', z' are given by
I 7)1 n a /3 y
I = af-itf ~^z'~ -(x'2 + y'* + z2) = x' + iy' = ^z~' *'
all of these lines are therefore lines of the linear complex
y = n.
To the point x, y, z there will correspond in space x', y', zf
the straight line whose coordinates are given by
I m n
i(x2—l)~~x2+l" — 2 ix
a $ y
txz xz . %z
y -j—,-iv z2—r-xy
x2-l J x2-l J x2-l
This straight line is such that
l2 + m2 + n2 = 0,
and therefore to x, y, z there corresponds in the other space
a minimum straight line.
It will be noticed that, in order to find what corresponds to
a point M2, it is only necessary to make use of the coordinates
of the point and the generating equations. In order to find
what corresponds to the surface M2 given by
Ix + my + nz + k = 0,
we must form the other Pfaffian equations of this M2 viz.
l + np = 0, m + nq = 0.
From the equations of the contact transformation we now
have (l) l+m(af-iy')-ntif= 0.
Eliminating x and y from the generating equations and the
262 TRANSFORMATION OF [202
equation of the given plane, we see that (on account of (1))
z also disappears, and we get
n (x' + iy') + mz' — & = 0.
The plane therefore has as its correspondent the minimum
liue n (x' + iy') + mz'-k=0, l + m(x'- iy') -nz'=0;
that is, has the same correspondent as the point
m —I — k
x=—, y = — , z = — •
§ 203. We next find what will correspond to the straight
une (i) a = mz-ny, /3 = nx-lz, y = ly-mx,
of which the coordinates are I, m, n, a, /3, y.
Eliminating x, y, z from two of these equations (there are
of course only two unconnected ones) and the generating
equations, we clearly get the generating equation of the M2
we require ; it is
(2) I (x2 + y'2 + z'2) -(3(x-iy) -m(x' + iy') + (n + y) z'-a = 0.
To find the minimum straight line, which corresponds to
any point on the given line I, m, n, a, /3, y, we must substitute
in the generating equations for y and z their values in terms
of x ; we get
x (lz' + n) = p — I (x' + iy),
x(l(x'—iy')—'m) = y + lz'.
Eliminating x from these two equations, we get the equation
of the sphere which corresponds to the given straight line ;
and one set of generators on this sphere consists of the minimum
lines which correspond to points on the given Une.
Writing the equation of the sphere in the form
(3) x'2 + y'2+z'2 + 2gx' + 2fy'+2hz' + c = 0,
and comparing with (2), we do not get unique values for the
coordinates of the straight line in terms of the coordinates
of the sphere. If we take r to be the radius of the sphere
(that is, Vf2 + g--¥hz — c taken positively), we see that there
are two straight lines in space x, y, z to each of which the
same sphere (3) will correspond.
These lines are respectively
I 7n 7b a /3 y
l" —g + if~~h — r~' —c" —g — if~h + r
204] STRAIGHT LINES INTO SPHERES 263
which we call the positive correspondent of the sphere, and
I m n a (3 y
1" —g + if" h + r" —c" —g—if~~h-r'
which we call the negative correspondent.
These two lines are conjugate with respect to the linear
complex y = n.
When r = 0, the sphere degenerates into a cone ; and any-
plane through the vertex is a tangent plane to the cone
(though of course an infinity of planes through the vertex
are tangent planes in a more special sense).
The two lines, the positive and negative correspondents
of the degenerate sphere, now coincide ; and therefore belong
to the linear complex y = n. This is another way of obtaining
the fundamental theorem, that a point in space %', y', z' has
as its correspondent in the other space a straight line of the
linear complex y = n.
By allowing /, g, h, c to increase indefinitely, without
altering their mutual ratios, we see that to the plane
2gx'+2fy' + 2hz' + c = 0,
there are two correspondents in space x, y, z, viz. the positive
correspondent
j _ rn ^
^g + V=h-Vh*+g*+/2
a (3 y
— c" —g—ifh+Vh^+g'^+f*
and the negative correspondent obtained by changing the
sign of the surd.
The straight lines therefore, which are perpendicular to the
axis of x, are not transformed into spheres, but into planes.
§ 204. Suppose now that we have the two spheres
xr2 + y/2 + z2+2g1x'+2f1y' + 2h1z, + c1 = 0,
x'2 + y'2 + z'i + 2g2x'+2f2y'+2h2z' + c2 = 0,
then, if llt mXi nx, al5 /31S yx,
are the line coordinates of their positive correspondents, we
have
lxa2 + l2ax^ -cx~c2, m1/32 + m2^1= 2g1g2 + 2f1f2i
Kiyz + Vi = 2A1£2-2r1r2,
264 SPHERES IN CONTACT [204
so that if the positive correspondents intersect,
2 9i92 + 2fif2 + 2hih = 2rir> + c1 + c2;
that is, the two spheres touch internally.
If the positive correspondents intersect so do the negative ;
for a positive and negative correspondent are conjugate to the
linear complex y = n.
If then two spheres touch internally the positive correspon-
dent of the first intersects the positive correspondent of the
second ; and the negative correspondents also intersect.
The two straight lines, the positive and negative correspon-
dents of a sphere, cannot intersect unless the sphere degenerates
into a point sphere ; for conjugate lines, with respect to a linear
complex, can only intersect when the lines belong to the
complex ; that is, when y = n, and therefore r = 0.
If the first positive correspondent intersects the second
negative correspondent, then the second positive correspondent
intersects the first negative correspondent, and the spheres
have external contact.
§ 205. If we are given a line whose six coordinates are
I, m, n, a, }3, y,
how are we to decide whether it is a positive or a negative
correspondent to the sphere to which it corresponds — for we
know there is only one such sphere ?
We always suppose the radius of the sphere to be positive,
and therefore by the formula
taking, as we may, I to be positive, we know that the line is
a positive correspondent if y > n, and a negative if y < n.
If then we are given two interesting lines, there is no
ambiguity as to whether the corresponding spheres intersect
externally or internally ; the question is settled by the positions
of the line with regard to the axes of coordinates.
If we neglected this consideration we should arrive at
paradoxical results by this method of contact transformation.
Thus, if we are given two intersecting straight lines A, B, we
know that, if any other two straight lines C, D intersect them
both, then C, D must themselves intersect. It would therefore
appear to follow, from the theory of contact transformation
explained, that if two spheres touch one another, then any
other pair of spheres, which touch both of the first pair, must
206] DUPIN'S CYCLIDE 265
also touch one another, a result which is obviously absurd.
To see where the error has arisen in the application of the
contact principle, suppose that the first two spheres touch
externally ; then A and B must be taken to be, one a positive,
and the other a negative correspondent of its sphere. We
suppose G to be a positive correspondent to its sphere C",
A a positive correspondent to its sphere A', and B a negative
correspondent to B' ; we now have C touching A' internally
and R externally ; and the only way this could happen would
be by C touching the two spheres, at their common point
of contact. Similarly D' must touch at this point ; and there-
fore C and D' do touch one another, but they are not any
spheres touching both A! and Bl '.
§ 206. The cyclide of Dupin is the envelope of a sphere
which touches three given spheres (Salmon, Geometry of Three
Dimensions, p. 535), there being four distinct cy elides, corre-
sponding to the different kinds of contact of the variable
sphere with the three given spheres A, B, G.
The four cases are when the variable sphere touches,
(1) A, B, C all externally or all internally ; (2) B, C externally
and A internally or B, C internally and A externally; (3)
C, A externally and B internally, or G, A, internally and
B externally ; (4) A, B externally and C internally or A, B
internally and C externally.
We shall only consider the first of these cyclides ; taking
a, b, c, d to be the positive and — a, — b, —c, — d to be the
negative correspondents of A, B, C, D we see that, either d
intersects a, b, c, or else it intersects the three negative cor-
respondents — a, —b, — c ; in either case it generates a surface
of the second degree.
A cyclide of Dupin in space x', y\ z' therefore generally
corresponds to a quadric in space x, y, z. If we take any
generator of this quadric and regard it as the generating
curve of a curve M2 in space x, y, z, its correspondent in the
other space will be a sphere touching the cyclide along a curve.
This curve must be a line of curvature on the cyclide ; for the
normals to the sphere along this curve intersect, and therefore
the normals to the cyclide along this curve intersect.
If, however, instead of regarding the generator of the quadric
as a curve M2 of x, y', z', we regard it as an ifj of elements of
the quadric ; that is, if we take the single infinity of elements,
consisting of the points of the generator and the tangent planes
at these points to the quadric, then the corresponding M1
266 LINES OF INFLECTION TRANSFORMED [206
in space x', y, z' is the line of curvature, with the tangent
planes at each point of it to the cyclide.
§ 207. Any surface in space x, y, z has at every point on it
two inflectional tangents. The surface therefore which corre-
sponds in space x , y' , z' will have, as corresponding to these
two inflectional tangents, two spheres each having contact
with the surface at two consecutive points ; that is, the
correspondents of the inflectional tangents will be the two
spheres whose radii are the principal radii of curvature
(Salmon, ibid., p. 264).
It will be noticed that any straight line drawn through a
point on a surface, and in the tangent plane, will be trans-
formed into a sphere touching the corresponding surface.
The peculiar property, however, of an inflectional tangent is
that it is a straight line through two consecutive points of
a surface, and also in the two consecutive tangent planes
at these points. It is therefore transformed into a sphere
having two consecutive elements common with the new
surface ; that is, it is a sphere whose radius is equal to one of
the principal radii of curvature.
By this contact transformation therefore the curves, whose
tangents are the inflectional tangents to the surface at the
point, are transformed so as to become the lines of curvature
on the surface in space x', y', zf.
If a surface has any straight line altogether contained in it
the corresponding surface will have a line of curvature, with
the same radius and centre of curvature all along this line.
§ 208. In general a quadric in space x, y, z is transformed
into a cyclide; but we shall now see that some quadrics are
transformed into straight lines in space x', y' , z'.
Let a = mz' — ny', fi = nx'—lz', y = ly' '—nix'
be a straight line in space x' , y', z' ; from the generating
equations we obtain, by eliminating x' , y' , z' ,
x((ai + (3)x — ny + (mi — l)z — 2yi) = (l + mi)y + nz + ai — (3.
This quadric therefore, instead of having a cyclide corre-
sponding to it in space x', y', z', has the line whose coordinates
are 7 a
I, m, n, a, j3, y.
It may be verified without difficulty that one system of
generators of this quadric belongs to the complex 1 = 0, and
the other to the complex y = n.
209] INTO LINES OF CURVATURE 267
§ 209. If we have a system of concentric spheres in space
x', y\ z, viz.
a/a + t/2 + z'2 + 2gx'+ 2f\j + 2hz' + c = 0,
where c varies, the corresponding system of manifolds in space
x, y, z will be straight lines satisfying the three linear com-
Plexe3 I m n + y _ J3
I" -g + if~ 2h ' -g-if
Two different manifolds will correspond to a given sphere
of radius r ; there will be the positive correspondent obtained
by making the coordinates of the straight line also satisfy the
linear complex 2 rl = y — n
and the negative by making the coordinates satisfy the
complex 2rl = n-y.
All these lines are generators of the same system on the
hyperboloid
C1) (if-g)x2-xy + 2hx-z + if+g = 0.
The generators of the other (the second) system on (1) are
x = t, z + ty = if+g+2ht + (if-g)t2;
the six coordinates of any one of these generators are
I m n a /3 y
0 = T ~ ^t ~ if+g + 2ht + (if-g) t2 ='~^t2='' -t'
Since I = 0, to each of these generators there will correspond
in space x', y', z/ a plane touching all the concentric spheres ;
these planes must therefore be tangent planes to the asymp-
totic cone ^ + gf + {y, +/)2 + ^ + h)2 = Q .
this result may be at once directly verified.
It may be noticed that all generators of the second system
belong to both the linear complexes
1 = 0 and y = n.
The hyperboloid (1) is given when we are given a gene-
rator of its first system ; one such hyperboloid can be
described through any straight line. We see therefore how
to construct the system of lines which will be transformed
into concentric spheres ; describe an hyperboloid of the form
(1) through any line ; then the lines, which will be trans-
formed to concentric spheres, are the infinity of generators
268 A SYMMETRICAL [209
of the same system as the given line. In particular that
generator, which belongs to the linear complex y = n, will
correspond to the centre of the given system of spheres.
§ 210. If a quadric is such that all generators of one system
belong to the linear complex y = n, then its correspondent in
space x', y\ zf, instead of being a cyclide, is a circle. For we
have, in space x, y, z, a system of generators intersecting two
fixed generators, and belonging to the complex y = n\ in the
corresponding figure therefore we must have a system of
points common to two spheres, that is, a circle.
§ 211. We now pass on to consider the more general case
of the two bilinear generating equations, when the four points
in space x', y', z\ for which the generating equations become
coincident, are not coplanar. We take these four points as
the vertices of a tetrahedron ; and we do not consider the
special cases which might arise, owing to two or more of
these vertices coinciding. We choose our coordinate axes so
that this tetrahedron has for its vertices the points
(0, 0, 0), (co, 0, 0), (0, co, 0), (0, 0, oo) ;
we thus have from the definition of the tetrahedron (employ-
ing the same notation as in § 199)
Ol __ Og __ Og _ OI4 __ ^i==&=-&==&==X
ax a2 a3 a4 15 bx b2 b3 64 2'
ft — ft — ft — ft — A ^1-^. = ^ = -* = a
ci " c2 "" c3 " c4 "" 3' d1" d2" d3" ^
We then take
-jr _ axx + a2y + a3z + a4 y _ fe1a; + 62y + 63g + 64
dxx + d2y + dzz + di> dxx + d2y + d3z + c/4 '
z ^x + ^y + ^z + c^
dxx + d2y + d3z + di
and thus see that by projective transformation the generating
equations may be thrown into the forms
axx' + byy' + czz' + d = 0,
xx' + yy' + zz' + 1 = 0.
If we keep x', y', z' fixed, these are the equations of two
planes, and therefore to a point xf, y', z' there corresponds
212] CONTACT TRANSFORMATION 269
a straight line in space x, y, z. The six coordinates of this
line satisfy the equation
la m/3 ny
(b-c)(a-d) (c-a)(b-d) (a-b)(c-d)'
that is, the line belongs to a complex of the second degree.
It can be at once verified that every straight line of this
complex is divided in a constant anharmonic ratio by the
coordinate planes and the plane at infinity ; on account of
this property the complex is called a tetrahedral complex.
We may look on the generating equations as the polar
planes of x', y', z, with respect to two quadrics, which do not
touch; the quadrics are referred to their common self-con-
jugate tetrahedron, viz. the coordinate planes and the plane at
infinity, and the polar planes intersect in a line of a tetra-
hedral complex of this tetrahedron.
In order to complete the contact transformation we must
add to the generating equations the three equations obtained
by eliminating A from
_ (\ + a)af _ (\ + b)y'
~P~ (k + c)V' q~ {K + c)zfi
_ ,_ (k + a)x ,_ (A + 6)y
p" (k + c)z3 q" (k + c)z3
that is,
p(b — c)z'y' + q(c — a)z'x' — {a— b)x'y'= 0,
rpzfx~'p'zx'— 0, qz'y — q'zy'= 0.
The equation p'(b —c)zy + q'(c — a) z'x' —(a — b)xy = 0ia con-
nected with these, and is not therefore an additional equation.
In this contact transformation the two spaces are symmetri-
cally related ; thus a point in either corresponds to a line of
the tetrahedral complex in the other.
§ 212. We must now find what corresponds in space x, y, z
to the plane 7 , , , 7
r Ix + my + n z + k = 0 .
Forming the equations of the Pfaffian system of which this
plane is the generating surface we have
l + np'= 0, m + nq'= 0,
and substituting for pf and q' in the equation
p' (b— c)zy + q' (c— a)zx — (a— b) xy = 0
270 TRANSFORMATION OF A PLANE [212 H<
tn
ml
fine
of the contact transformation we have
(1) l(b — c)yz + ni(c — a)zx + n(a — b)xy = 0.
This, however, is not the only generating equation defining
the M2 which will correspond to the plane in the other space.
For, eliminating y' , z' from
axx' + byyf + czz' + d = 0,
xx' + yy'+ zz'+\ = 0,
lx' + rtiy' + nz' ' +k = 0,
we see that by aid of (1) x' disappears at the same time, and
therefore all the three-rowed determinants vanish in the matrix
ax, by, cz, d
(2) x, y, z, 1
I, m, n, k
These are the equations of a twisted cubic, viz. the locus
of a point whose polar planes with respect to the quadrics
#2 + y2 + z2 + i — 0 and ax2 + lyi + Cz2 + d= 0
intersect on the plane
Ix + my + nz + k = 0.
This cubic passes through the origin and the points at infinity
on the axes of coordinates.
To a plane in one space there will then correspond in the
other space the twisted cubic given by the above equations.
As a, b, c, d are fixed, when the contact transformation is
fixed, we may call I : m : n : k the coordinates of this twisted
cubic.
VIZ,
COOT
inn
piau
§ 213. The coordinates of any point on this cubic are
Clli'il
iauiii
skill
the i
tetral
I (t + d) m(t + d) n(t + d)
k(t + a) * k(t + b) k(t + c)
Since therefore the six coordinates of the line in space
x', y', z' which corresponds to x, y, z are
V = (b — c)yz, im!= (c—ajzx, n'= (a—b) xy,
a'=(a-d)x, (S' = (b-d)y, y' = (c-d)z,
the coordinates of the line which corresponds to a point on
the twisted cubic are p
l'=z (b — c)mn(t + a) (t + d), a'= (a — d)lk(t + b) (t + c),
with similar expressions for the other coordinates.
214] INTO A TWISTED CUBIC 271
The coordinates of the line joining two points on this
twisted cubic are
, _ l(a-d) (<! — £2) , _ mn (b - c) {tY - t2) (tt + d) (t2 + d)
~k{t1 + a)(t2 + a)' k2(t1 + b)(t2 + b)(t1 + c)(t2 + c) '
with similar expressions for m', n', fi', y ; such a line there-
fore belongs to the tetrahedral complex
Va! m'tf n'y'
(b — c)(a — d) (c — a)(b — d) (a — b)(c — d)
and so is divided in a constant ratio by the coordinate planes,
and has, as its correspondent in space x\ y', z\ a point on the
plane 7 , . , .
r Ix + my + nz + k = 0.
The twisted cubic which in one space corresponds to any-
plane in the other always passes through four fixed points,
viz. the origin and the points at infinity on the axes of
coordinates ; and any straight line which intersects the cubic
in two points is divided in a constant ratio by the coordinate
planes. This ratio does not depend on the position of the
plane which corresponds to the cubic.
It is generally true that any straight line intersecting any
twisted cubic in two points is divided in a constant anhar-
monic ratio by the faces of any tetrahedron inscribed in the
cubic. In order that a twisted cubic may belong to the
family we are here considering it is only necessary that it
should pass through the origin and the points at infinity on
the axes and be such that the anharmonic ratio for this
tetrahedron has the assigned value which defines the tetra-
hedral complex. We shall speak of these cubics as cubics of
the given complex.
Since a plane can be drawn to pass through any three points
we see that a twisted cubic can be drawn to intersect any
three lines of the tetrahedral complex ; for a line of this
complex corresponds to a point in the other space.
§ 214. We next find what corresponds to the line
(1) a = mz' — ny', /3 = nx' — lz', y = ly' — mx'.
Eliminating y' and sr from the equations of this line and
the given generating equations of the contact transformation,
axx + byyf + czz' + d - 0, xx' + yy' + zz' + 1 = 0,
272 A GEOMETRICAL THEOREM [214
we get
^ ' x'(alx + bmy + CTiz) + dl+byy — c(3z = 0.
These are the equations of a generator of one system on
the quadric
(3) a(b—c)yz + (3(c—a)zx + y(a—b)xy
+ 1 (a — d)x + m (b — d)y + n (c — d)z = 0;
and since (2) corresponds to x\ y', z' we see that this system
(the first system, we shall call it) of generators on this quadric
belongs to the tetrahedral complex.
Now any quadric passing through the origin and the points
at infinity on the axes of coordinates is of the form (3) ; we
thus have the following interesting theorem in geometry :
the generators of a quadric are divided in a constant anhar-
monic ratio by the four planes of any inscribed tetrahedron *.
The following is an analytical proof not depending on
contact transformation theory. The equation of the quadric
referred to the tetrahedron as tetrahedron of reference is
ax yz + bxzx + cx xy + axw + byw + czw = 0.
The conditions that the line
ly—mx = yw, nx — lz = j3w
may lie wholly on the quadric are
a1mn + b1nl + c1lm = 0, a-^^y—bly + cl^ = 0,
ai(ny—m/3) + l {cxy — 6x/3) +1 (la + mb + nc) = 0.
Eliminating I from these equations we get
(cj m2 j3 — b± n2 y) (cx m + bxn)
+ win (cx bm2 + bx en2 + (c^ + bb± — aax) mn) = 0,
mn(cft — by) = /3y (Cjm + ^w).
These equations give us to determine the ratio of /3 to y
bx bn2 y2 + cx cm2 /32 + (b1b + c1c — a1 a) mnfty = 0 ;
and we have similar equations for a : /3 and a : y.
If the straight line intersects the faces of the tetrahedron
* This and much more about the tetrahedral complex will be found in
Beruhrungstrans/ormationen, Lie-Scheffers, Chap. VIII.
216] DEGENERATION OF THE QUADRIC 273
of reference in A , B, G, D respectively, and if the anharmonic
ratio . * „~ is denoted by A, we therefore have
AD .BG J
ax a A2 — (ttj a + bx b — ct c) A + bx b = 0,
so that the generator is divided in a constant ratio by the
faces of the tetrahedron of reference.
§ 215. There are two systems of generators in the quadric
(1) a(b — c)yz + fi(c — a)zx + y(a — b)xy
+ 1 (a —d) x + m (b —d) y + n (c — d)z = 0.
To the first system of these generators we have seen that
there correspond, in space x\ y', z\ the points on the lines
(2) a = mz'— ny\ /3 = nx' — lz\ y = ly' — mx'.
The equations of the generators of the other system are
t (Ix + my + nz) + alx + bray + cnz = 0,
t (l—(3z + yy) + byy—cfiz + Id.
The six coordinates of this generator are given by
a'=l(a + t), p'=m(b + t), y=n(c + t),
„_a(b + t)(c + t) „,_P(c + t)(a + t) y(a + t)(b + t).
1 ~ dTt ' m~" d+t > n~- d+t
and therefore to any generator of this system there cor-
responds in space x', yf, z' the quadric
(3) o'(6 - c) y'zf + (3'(c-a) z*x' + /(a - b) x'y'
+ V(a- d) x' + m'(b — d) y' + n'(c -d)z'=Q.
Since all generators of the first system intersect each
generator of the second, we can conclude that all points lying
on (2) must also lie on (3) ; that is, (3) contains the line (2) ;
this may easily be verified directly.
§ 216. If the straight line whose coordinates are
I, m, n, a, /3, y
belongs to the tetrahedral complex, that is, if
la ra/3 ny
(b-c)(a-d)" (c-a)(b-d) (a-b)(c-d)
the quadric of the form (1) of § 215 which corresponds to
the line is a cone.
CAMPBELL
274 ILLUSTRATIVE EXAMPLES [216
The go2 of elements which consists of points on the above
line, together with the infinity of planes which contains the
line, is therefore transformed into the cone M2.
We know, however, that the M2 which corresponds to a
line of the tetrahedral complex is a point M2, so that this
point M2 must coincide with the cone M2. There is of course
nothing paradoxical in this ; for the point must be the vertex
of the cone, and any plane through the vertex will be a
tangent plane to the cone.
The quadric which corresponds to a straight line has, like
the twisted cubic which corresponded to the plane, the pro-
perties of passing through the origin and the points at infinity
on the axes of coordinates ; it has also the property that its
generators of one system are divided in the assigned ratio
which defines the tetrahedral complex. We shall call any
quadric of this family a quadric of the given complex.
The contact transformation we have now considered has
the property of transforming point M2's into the M2's of lines
of the tetrahedral complex ; or, as we may briefly express it,
points into lines of the complex. It also transforms planes
into twisted cubics of the complex ; and straight lines
generally into quadrics of the complex, though, if the line
belongs to the complex, the quadrics degenerate into points.
§ 217. We may now apply this method of transformation
to deduce new theorems from theorems already known.
Thus a straight line can be drawn through any two points
in space ; therefore a quadric of the complex can be drawn
through any two lines of the complex.
Again any two planes intersect in a straight line ; therefore
a quadric of the complex can be drawn through any two
twisted cubics of the complex.
A straight line in space which intersects three fixed lines
intersects an infinity of other fixed lines ; therefore a quadric
of the complex which touches three fixed quadrics of the
complex touches also an infinity of fixed quadrics of the
complex.
One more illustration of the method will be afforded by
taking any six points Px, P2, P3, P4, P5, P6 on a twisted
cubic of the complex ; to these six points will correspond
six lines of the complex, and all of these lines will lie on
the plane which corresponds to the cubic. These lines are
divided in a constant anharmonic ratio by the coordinate
planes and the plane at infinity ; and therefore are divided
in a constant ratio by the sides of a fixed triangle. They
218] ON CONTACT TRANSFORMATION 275
therefore all touch a parabola ; let AB correspond to Pls BO
to P2 and so on ; B will then correspond to Pj P2 . If we
now apply Brianchon's theorem to the hexagon ABGDEF
formed by the six lines, we see that AD, BE, and CF are
concurrent. To AD will correspond the quadric of the
complex which contains the lines Px P6 and P3 P4 ; to BE
the quadric with the generators Px P2 and P4 P5 ; to CP the
quadric with the generators P2 P3 and P5 P6 ; the theorem
which we can now deduce from Brianchon's is that these three
quadrics have a common generator.
§ 218. We have now examined the first two classes of contact
transformations and there remains the case where there are
three generating equations ; but as we can now express
x', y', z in terms of x, y, z, and x, y, z in terms of x', y', z' ,
this is a mere extended point transformation. We have had
examples of this class of contact transformation in Chapter II,
and shall return to the subject in Chapter XX on differential
invariants, so that we need not now consider it further.
T 2
CHAPTER XVIII
INFINITESIMAL CONTACT TRANSFORMATIONS
§ 219. If zt xlt ..., xnipx, .»,])„ are the coordinates of an
element in w-way space,
z = z + tQix^, ..., Xn,z,p1, •",pn)>
X$ = X^-\- Iqi (iCj, ..., Xn, Z, 2^1) •••» Pn)> V* == *' •••» ^7>
Pi = Pi + *»< (#1> ..;Xn,Z,p1...,pn)
is an infinitesimal transformation of the elements, if t is a
constant so small that its square may be neglected.
The transformation is an infinitesimal contact transforma-
tion if the Pfaffian equation
dz— p1dx1 — ...— pndxn = 0
is unaltered ; that is, if we have
i = n i = n
dz'-^, p'i dx'i =(l+pt)(dz-^pi dx(),
where p is some function of the coordinates of the element.
Now dz'= dz + td(, dx'i = dx$ + td£it dp\ = dpi + td^ ;
i = n
if then we take W= 2 Pi €i — C>
we have
i=n i=n i=n i=n
dz' - 2 Pi dx'i = dz-^ p( dXi + t(d(-^p{ d^ - 2 *i dx{)
i = n i = n
= dz-^ Pi dx{ -tdW + i2te dPi—^i dx{)
(neglecting small quantities of the order t2) ; and therefore
i = n i — n
2 (f » dPi - *i dxi) -dW= P{dz-^pi dx{),
221] THE CHARACTERISTIC FUNCTION 277
, 2>TF 3 IT 3 IT SIT
so that 6=^ p=— ^> ^ = -^-^us
i=n i=n
§ 220. Conversely if W is any function whatever of the
coordinates of an element,
, v , 3If , X,W 3W\
(1) Xt = Xi + t — , Pi = Pi-t(— +Pi— ),
t=»i
will be an infinitesimal contact transformation ; for
i = n
dzf — 2 Pi dxi
= dz — ^lPidxi
= (i -*-^) (^-2^ <&*) •
The function If is called the characteristic function of the
infinitesimal contact transformation; and the corresponding
infinitesimal operator is
If If does not contain z, and is homogeneous of the first
degree in Pl, ...,_£>„, the infinitesimal contact transformation is
a homogeneous one.
§ 221. Suppose now that (/> (z, xx, ...,xn> plt ...iPn) is any
function of the coordinates of an element, then zf,x[, ...,x'n,
278 THE CHARACTERISTIC MANIFOLDS [221
p[,...,p'n being the contiguous element defined by (1) of
§ 220,
4, (*>;,...,<, p[, ...,p'H) = <t> + t[W,<b]-tW^,
where
The necessary and sufficient condition therefore that the
function <b should admit the infinitesimal contact transforma-
tion with the characteristic function W is
[W,<t,] = W3£.
Similarly we see that the equation <b = 0 admits the con-
tact transformation if the equation [ W, #] — W -r- = 0 is
connected with (b — 0.
If the equation <b = 0 admits the contact transformation,
with the characteristic function W, the equations W = 0 and
0 = 0 will be equations in involution.
§ 222. If <b1 = 0,...,<hm = 0, I
are any m equations in involution (§ 153), then, W = 0 being
any equation connected with the system, this system will
admit the contact transformation, whose characteristic function
is If.
If we are given any function <f>(z,x1»...itonfp1, ...,pn) of
the coordinates of an element, we can find 2n unconnected
functions in involution with this function ; let these func-
tions be
(p1^Z,X1, ..., &n,Pi> '••iPn)> '••) 9 2n \?i *^1» •••» ^n> Pi* %t,*Pn) '
it will now be proved that the equations
( V 9i \z> ®l > • • • » ®n > Pi ' •••' Pn) = ri \z >xl> •*«»«%» Pi* "•iPn)>
(i = 1, ...,2%),
define a simple infinity of united elements, that is, an Mt
containing the assigned element z°, x\, ..., xQn , p\, ...,pn.
and x1 + dx1, ...,xn + dxn, z + dz, p1 + dp1,...,pn-\-dpn
222] OF A FUNCTION 279
are two consecutive elements satisfying the equations (1)
then
and since all the functions 01S ..., <f>2n are in involution with 0
we must have
l- — n
There are 2?i equations of the form (2) by means of which
we can determine the ratios of
the equations (3) to determine the ratios of
jfc = 71
30 30 ^ 30 30 30 30 30
*Pl *Pn *Ph *®1 *3 *Xn ln7)Z
are exactly the same ; and therefore we conclude that
dxx dxn dz
(4)
H H_ k = n 30_
tyl *Pn 2* Pk )pk
dpi dPn
30 30 30 30
~~ 3^ ~Pl lz~ ~ lx~n ~Pn 3l
Since the equations (4) satisfy Pfaff's equation
dz = p1dx1 + ... +pndxn,
we conclude that the infinity of elements satisfying each of
the equations (1) consists of united elements.
Any simple infinity of elements satisfying the equations (4)
is called a characteristic manifold or Mx of the function 0.
It is possible to describe one, and only one, of these
characteristic M^a through any assigned element of space
z°, x\, ...,#£, Pi, ..*,Pn ; and it is easily seen to lie altogether
on the manifold
¥ \Z> **i> •••> *^tt> Pi* "'iPn) == * \P > *^i' •••» *t»> Pl> '"'Pri/i
as well as on each of the manifolds given by (1).
280 LINEAR ELEMENTS [222
Wc shall now prove that by any contact transformation
a characteristic Mx of a function is transformed into a charac-
teristic Mx of the corresponding function. This follows at
once from the facts: (1) that two functions in involution
are transformed into two functions in involution ; and (2)
that the characteristic Mx of a function <\>, which contains the
element z°, x\, ...,a?°, p\, >>>,Pn, consists of all elements com-
mon to
(i = 1, ..., 2ri),
where 4>x, ..-9to» are an7 2w unconnected functions in involu-
tion with (p.
§ 223. We may now interpret an infinitesimal contact
transformation as follows: take any element z, x1} ..., xn>
p ...,pn and construct the characteristic Mx of the character-
istic function W which contains this element. Imagine an
element to be moving along this M1} the consecutive element
to the one assumed is
z + tz, x1 + txli...,xn + txn, p1 + tp1,...,pn + tpn,
where t is the small interval of time taken to move to this
consecutive position ; the infinitesimal contact transformation
which corresponds to IT is then given by
z'= z + t(, x[ = xx + t£x, ...,x'n = xn+tgn,
P'l = Pl + t*l> —>Pn= Pn + t**'
where
xx = £15 ...,xn = in, 2h = tt1, ...,pn = 7tn, but z-W = C
We may then say that the velocity of an element, under the
effect of the infinitesimal contact transformation whose
characteristic is W, is composed of a velocity along the
characteristic M1 of W containing this element, and a velocity
along the axis of z; the ratio of the z component of the
first velocity to that of the second being as
2^ to -w.
§ 224. If P and P' are two consecutive points in space, the
straight line joining the points and terminated by them is
called a linear element.
224] AND INTEGRAL CONES 281
If we take any point z, xx, ..., xn then cou_1 elements
z, xx, ...,xn, px, ...,p>n pass through this point, and satisfy the
equation <£ = 0 ; it therefore follows that con-1 characteristic
ilf/s of this equation pass through any point. Taking
to be the coordinates of the linear element joining z,xx, ...,xn
to a consecutive point on any one of these characteristic Mxs,
we see that these coordinates must satisfy the equation (or
equations) obtained by eliminating px^...,pn from the equa-
tions
Pi Y^T "•"••* ~^~Pr
*Pl *Ih *Pn *Pl n *P
n
This equation is called the equation of the elementally
integral cone of (f> = 0 at the point xx, ...,xn,z.
We have seen that if the equation <£ = 0 is transformed by
a contact transformation into \j/ = 0, then the characteristic
31xs of (f) = 0 are transformed so as to be the characteristic
Mx's of y\r = 0. It does not, however, follow that the elemen-
tary integral cones of 4> = ° will be transformed into the
elementary integral cones of ^ = 0 ; for characteristic Mxs,
meeting in a point, will not in general be transformed to
characteristic Mxs, meeting in a point.
If, however, the transformation is merely a point transfor-
mation, the elementary integral cones of one equation will be
transformed to the elementary integral cones of the other.
In particular, the point transformations which leave a given
equation of the first order unaltered, will also leave the
system of integral cones unaltered, though naturally these
cones will be transformed inter se.
Looking on
p1dx1 + ... +pndxn = dz
as the equation of an elementary plane whose coordinates
are px, ...,pn, we easily prove that 0 = 0 is the tangential
equation of the elementary integral cone of <fi = 0 at the
point z,xx, ...,xn.
Conversely, suppose we are given an equation, homogeneous
in dz, dxx , ..., dxn , and connecting z,xx, ...,xn, dz, dxx , ..., dxn ,
the coordinates of a linear element ; then, if, regarding
dz:dx1:dx2: ... as the variables, we find its tangential equa-
tion, we shall have a differential equation of the first order,
282 MONGIAN EQUATIONS AND [224
of which the given equation will be an elementary integral
cone.
We thus see that any point transformation, which leaves
a differential equation of the first order unaltered, will also
leave unaltered an equation between the coordinates of a linear
element ; and, conversely, a point transformation, which
leaves an equation between the coordinates of a linear element
unaltered, will also leave unaltered a differential equation of
the first order.
An equation between the coordinates of a linear element
is called a Mongian equation. We have now proved that
to every Mongian equation there will correspond in general
one differential equation of the first order ; and conversely
to every differential equation of the first order there will in
general correspond a Mongian equation.
We say, ' in general,' because, for instance, if the elementary
integral cone at a point shrinks into a line (as it would if the
given differential equation were linear) there would not be
one definite Mongian equation but the several equations
which make up the line; and other cases might arise where
the result of eliminating p1} ...,pn from (1) would be several
equations.
So also if the Mongian equation were linear in dz, dxx, . . ., dxn
instead of having one equation between the coordinates
z, xlf ...,xn3 px, ...,pn, we should have n such equations ; for
the envelope of a plane touching a given plane is the plane
itself.
§ 225. Example. We saw in § 33 that the point transfor-
mations which were admitted by
l+pz + q* = 0j
were also admitted by
dx2 + dy2 + dz2 = 0,
the equation satisfied by the linear element of a minimum
curve ; these two equations are clearly associated in the
manner just described.
A straight line of the tetrahedral complex which we con-
sidered in Chapter XVII has its linear elements connected by
the equation,
(b—c)(a—d)xdydz + (c — a)(b—d)ydzdx
+ (a—b)(c — d)zdxdy = Q.
225] DIFFERENTIAL EQUATIONS 283
If we form the associated partial differential equation, by-
expressing the condition that
pdx + qdy = dz
may, when we substitute pdx + qdy for dz in (1), lead to
a quadratic with equal roots in dx : dy, we obtain
. . (px (a—d) (b—c) + qy (b—d) (c—a) + (c — d)(a-b))2
= 4pqxy(a — d)(b — c) (b — d) (c — a),
which may also be written in the form
</px(a-d)(b-c) + Vqy(b-d)(c-a)+ V(c-d)(b-a) = 0.
We could now find the group — assuming such to exist — of
point transformations admitted by (1), and the group admitted
by (2) ; and seeing that these coincide we should verify the
general theorem of their coincidence.
Without, however, actually finding either of these groups,
we may easily verify that the point transformation
y' z'
+
X = e^c-a) (b~d) V(.<*-b) (c-d^
d x'
(3) y = eV(a-6) (c-d) V(fc-c) (a-cO
x' \f
Z = eV^b~c) (a_d) V(c-a) (6-cO
transforms
(b—c) (a—d)xdydz + (c—a)(b—d)ydzdx
+ (a — b) (c — d)zdxdy = 0
into dx'2 + dy'2 + dz'2 = 0.
The group found in Chapter II will therefore, when the
transformation (3) is applied to it, be a group transforming
any linear element of a tetrahedral complex into another such
linear element ; and will therefore leave unaltered the equa-
tion (1). It may also be easily verified that (3) will transform
(2)int0 l+^+?,2 = 0.
We can always find a contact transformation which will
transform any given partial differential equation into any
other assigned equation, if both are of the first order; this
we have proved in § 183 ; but it is not generally true that
we can find a point transformation which will do so. The
284 CHARACTERISTIC FUNCTION OF [225
example which we have just considered, suggests that if we
wish to determine whether two assigned equations can be
transformed, the one into the other, by a point transformation,
it may be more convenient to determine whether or no the
corresponding Mongian equations are transformable into one
another by a point transformation.
§ 226. Let W denote the infinitesimal operator which
corresponds to the characteristic function W, viz.
As we vary the characteristic function we get different
operators ; we must now find the alternant of two such
operators.
To do this, we take
yx= xlf ...,yn = xn, yn+1 = z, |r
n — & v — 2«_ Tj — _a -ur
P\ — n >-">2Jn— n > J~L — (in+i 'v >
Hn+1 !/b+1
and we find the operator in the variables
2/i» ■••> Vn+i> ?i> •••> 9Wi>
which has the same effect on any function of these variables
W
(provided that it is homogeneous and of zero degree) as
the operator W has on the same function expressed in terms jD ; t
01 x-^ , . . . , xn , z, p1 , . . . , pn .
Let the function on which we are to operate be
9 \xX) ...,xn, z, p±, ...,pn) = y (y1, ..., yn+1, (Zu •••>2fw+i),
then by § 184
7)(j) ^y ^4* ^Y ^4* ^Y
ty< *™lqx *x4 lVi *z-*yn+1
(i= 1, ...,n),
and, since y *s homogeneous of zero degree,
i = n i=w
2 0<h ^ <)\]s c)\b
*Pi ""Hi- qn+1*qn+i
226] THE ALTERNANT 285
We now get
i = n i = n
■777. "V ° /—ll\0\lr -^ 0 s— ll^d\lf
d (-H\ ty _ * (~H\ ^
, g H .
and therefore
^ *>& ty* ^ ty* *ft
where if is the infinitesima] homogeneous contact operator
which corresponds to the characteristic function H.
That is, W operating on any function of xv . . . , xn z, pv . . . , pn
has the same effect as H on the equivalent function of
Vi> •••>&+!, ?if»>?»+i where iT = -gn+1F.
It therefore follows that
r2 Tf2- Tf2 ^ = HXH2-H2HX = (Hx, H2).
We proved in § 184 that Wx and TT2 being any functions of
xx , , . . , #n , z, px , . . . , ^?n
and therefore
— r— (HVH2) =^-(qn+1Wx, qn+1W2)ytq
<> w
- q*n{WvWJ.M KW^yn+1 W*7>ynJ
That is, W 1 TT2 — TF2 TFX has the characteristic function
L^i> ™2\z^v ywx ^ vv2 ^^ )>
286 FINITE CONTINUOUS [227
§ 227. We next proceed to show how the operator W is
transformed by the contact transformation
(1) x'{=X{, z'=Z, p'i-Pi,
with the multiplier p defined by
dZ— 2 pi dXi = p(dz-^Pi dxi)-
Take
Q Q
xl = Vl> "■ixn = Vni z = 2/n+i' Pi = I--' '">Pn = ~Z~
Vn+l Hn+i
xl = V\ > • • • ' xn = 7/n* Z ~ an+1 » Pi = ~J " ' • • •' 9 = T7 5
and let j/J = J^ q'i = Qi, (i = I, ...,n + l)
be the homogeneous contact transformation equivalent to (1)
obtained by eliminating #, £> and #', p' from (1) and (2).
Let H = —qn+1W; let if' denote the function of y\ q' equi-
valent to H ; and let V be that function of x\ p' which is
given by K'= -£+lV.
We now have H = K' and therefore by § 1 83 H = K' ; and
having proved that W = H, and V — Kf, we conclude that
Now V— ^±± W = pW ; in order therefore to express W
q?i+i
in terms of the variables x[, ...,x'n, z', p[, ...,p'n we find p, and
then express pW in terms of these variables by (1) ; the func-
tion thus obtained will be the characteristic function, with
respect to the new variables, of the required operator, equi-
valent to W.
§ 228. The totality of contact transformations form a
group. For, z\ x[, ...,afn, p[, ..., p'n being the element derived
by any contact transformation from z, xt, ..., xn, px, ...,pn,
and z" , x", ...,x'n, p", ..^p'/t being similarly derived from
z', x[,...,x'n, p[,...,pfn by any other contact transformation,
we deduce from
i = n i = n
dz' - 2 p\ dx'i = p (dz - 2??i dxt) ,
i = n i = n
and te' - 2 Pi dx'J = p' (dz' - 2 Pi dx^,
228] CONTACT GROUPS 287
that dz" - 2 Pi dx'l = PP' {dz - 2 Pi <&?<)■
Therefore z" , x[' , ...,xn', p", ...,p'n' is derived from
Z, Xj, .,., xn , p j , . . . . pn
by a contact transformation ; that is, contact transformations
satisfy the definition of a group, and clearly, the group is
a continuous one.
We are now going to explain what is meant by a finite
continuous contact group ; it will be seen that many of the
properties of finite continuous point groups can be transferred
to the groups now about to be defined.
II Xj = A.£ [X^, ..., Xn, Z, Pj_, ...)^?nj #ij ..., Oir),
Pi = -*i v^i) •••> ^n ' ^' Pl> ' "> Pn> ^1' •"' ^r)>
Z =: Z (A3 j, ..., Xn, Z, Pj, ...,pn, ttj, ..., Olr)
is a contact transformation for all values of the constants
Oj, ...,ar ; and if from these equations and
®i = ■**■{ \p^ii • ••> &n> ^ ' .Pl> '">Pn> ^1» •••' ^r/>
.IPi = ■* j (^1 5 • • • 5 ^/j > 2 , ^?j , • . . , pn , Oj , . . . , Or^,
0 = Z (£CX , . . . , xn , z , Pi, ..., pn , t>i , . . . , br) ,
where blt ..., br are another set of constants, we can deduce
*t = -^-i v*^i ' • • • > *^n ' ^' ^l ' • • • ' Pn » ^i s • • • > ^r)>
Pi = -* i («^i j • • • j %n ' ziP\i'"iPni fn ■••)fr/'
# = Z (2^, ..., iCn, £, ^j, ...j^nj ^i> •••> Cf/j
where cl9 ...,cr are constants depending on als ...,ar, bx, ...,br,
then X^, Pi, Z are said to be functions defining a finite con-
tinuous contact transformation group.
Such a group will have r independent infinitesimal operators
Wi, ..., Wr. We see at once that the corresponding character-
istic functions must be independent, that is, there must be no
relation of the form
c1W1+...+crWr= o,
where c1,...,cr are constants, connecting the characteristic
functions. Also any finite transformations of the group can
be obtained by endless repetition of the proper infinitesimal
transformation.
288 EXTENDED POINT [228
The alternant of any two of these operators is not inde-
pendent of the set of operators ; we must therefore have
(ir#ir1)a2<ta*,„ (izllZ'*)-
Conversely, if we have r independent operators satisfying
these conditions, they generate a finite continuous contact
transformation group. If we use the symbol {TT4-,Wft} to
denote [Wi,Wh]gi3CtP-W1-^ +W2^, we can express
this fundamental theorem in terms of the characteristic
functions thus :
h = r
These theorems for contact groups follow at once from what
has been proved for point groups.
The constants cilih, . . . are still called the structure constants
of the group.
§ 229. If W is of the particular form
where £x, ..., £n, ( involve only xlf ...,xn,z, the corresponding
operator is said to be the extended operator of
and Z'= z + t((x1,...,xn,z)
xi = ^i + ^sz \xn — » ^to > z)
Pi = Pi + ^ "i \xl > •'•>xn'z> Pl> "•iPn)
is said to be the extended infinitesimal point transformation
01 X± = X± + t^j [X^y ..., Xn, Z), ..., Xn = Xn-f- tgn (#j, ..., Xn, Z),
z = z + tQyx-^, ...,xn, z),
and it is entirely given when the point transformation is
given.
Suppose that
^ = An (ajj, ..., xn, z, alt ..., &r), z = Z [Xi, ..., #M, 2, c&1} ..., <xr^
are the equations of a point group ; when we know the form
230] TRANSFORMATIONS 289
of the functions X15 ...,Xn,Z we can, as in § 185, find the
form of the functions Pj, ...,Pn where
It is now obvious that in the variables z,xx, ...,xn, 2h>-'-iPn
these (2 7i + 1) equations define a group of order r ; for, from (1)
and
(2j #4- = A^a^, .,., a??i , 0 , Pj, ..., Oj.j,
2 ^= ^ ^a?j , • • •, #^jj *j "jj • • •) ^j-/j v" ~" -I} • • •> 'vj
where b1,,..,br are constants, and where the equations (2)
involve the additional equations
Pi — -* i (#i ? • • • j ^ii j Z>Pi)'">Pn> ^1» •••»^n/> (& = 1, ..., Tlj,
we may deduce
Z = Z \pC\i •••} #wj #> C1} ..., Cr), (^ = *» •••! %/5
where c15 ...,cr are constants which are functions of the sets
av ...,ar, Oj, ...,or ; and from (3) we may deduce
_£>£ = Jr i \X^, • . .) ffini %iPi> •'•>Pn> *-i5,,,s^r/' \ ~ " *»'•••'»•»/«
§230. Let Tfj,..., TTr be the extended operators of this
group in the 2n+l variables, and Ux, ..., Ur the operators
of the original group ; it can now be proved that the structure
constants of the extended group are the same as the structure
constants of the original one.
h = r
Let (Wi,Wk) = ^ yilckWhi
j (Ui,Uk) = ^cihhUhi
and let Wi=Ui+Vi,
— 7> d d ,
so that in V,- the terms r — > ■••>; — j ^- do not occur.
* da^ dxn cz
We now have
I ={Ui,U:k) + operators in _,...,_ only,
for the coefficients of^ — >•••>- — s —in ^ and £7^ involve
^Xx dXn dz
oolyxll...,xniz.
CAMPBELL TJ
290 CONDITIONS FOR THE SIMILARITY [230
a
We have, therefore,
2 Vikh Wh = 2 cikh Uh + operators in ~ ^- only ;
so that
h = r }) <> 7)
2 (Yikh-Cikh) Uh = operators not involving ^-> ""^ ^*
It follows that y^ = cikh for all values of i,k,h; that is, the
extended group has the same structure constants as the
original point group.
We see, therefore, that if we are given any structure con-
stants, we can always find at least one contact group (viz. the
extended point group) with the assigned structure ; and,
therefore, the third fundamental theorem also holds for contact
transformation groups.
§ 231. We now proceed to obtain the necessary and
sufficient conditions that two groups of contact transformations,
in the same number of variables, may be transformable, the
one into the other, by a contact transformation. Since a con-
tact transformation in z, xl, ..., xn, plt ...,pn can be expressed
as a homogeneous transformation in y1, ..., yn+1, qt, ..., qn+u
it will be sufficient to consider this problem for the case of the
homogeneous contact groups.
Suppose Hlt ..., Hr are the r independent characteristic
functions of a finite continuous homogeneous group ; let us
apply any homogeneous transformation, and let these functions
become respectively K'x, ..., K'r when expressed in terms of
the new variables 2^» ...» 2/ro> ?'i> •••>#?* Dv ^ne given homo-
geneous contact transformation
Vi= P»(2/l>«"»2/n> ?l> •••> <ln)> q'i = Qi(yv~;yn> ?!>•••> <ln\
(i= l,...,m).
We know that (H{, Hj)y,q = (K'i>Kj)q',y'> an(^ therefore
h=r
(Ki,Kj)y,q'=^cijhK'h;
so that the new characteristic functions in y[, ...,y'n, q\, >",qrn,
generate a group with the same structure constants.
Now the functions H1, ..., Hr are independent in the sense
that there is no relation between them of the form
c1JB1 + ...+c,jrr = o,
232] OF CONTACT GROUPS 291
where c1, ...,cr are constants; but they do not need to bo
functionally unconnected. Suppose that H1, ..., Hm are func-
tionally unconnected, and that the other functions Hm+1, ..., Hr
can be expressed in terms of them, so that
Hm+t= &iM-*(-Si»—»-Sm)> (* = 1> ..-.r*— m),
and therefore
If then we are given the r characteristic functions of a
transformation group, viz. H1,...,Hr, and the r characteristic
functions of another group, viz. Kv ...,Kr, we cannot trans-
form the one group into the other, so that H^ may become Kif
unless the structure constants are the same, and unless the
functional relations are also the same.
§ 232. We shall now prove that these necessary conditions
are sufficient. Let Hv ...,Hr be the one independent set of
characteristic functions such that
h = r
and Hm+t = <f>m+t(H1,...,Hm), (t = 1, ...,r-m) ;
and let Kx, ..., Kr be another set of independent characteristic
functions such that
h=r
(Ki,Kj)=^cijhKh,
and Km+t = cf)m+t(K1,...,Km), (t = 1, ..., r-m).
JjTj, ..., Hm now form a homogeneous function system with
the structure functions iv^, ...,ivit ... where
ij — ^Lcijs^8^~ J-* ci,j,m + t<rm + t\**i> •••» -"m/> wi — *J
\7
By what we have proved in § 182 there can now be found
a homogeneous contact transformation, which will transform
H1,...,Hm into Kx, ..., Km respectively, since the two systems
have the same structure functions.
It is clear that this transformation will also transform
Hm+1, ...,Hr into Km+1, ..., Kr respectively; the necessary
conditions are therefore also sufficient conditions.
It might be supposed that we could from this theorem
u %
292 REDUCIBLE CONTACT GROUPS [232
deduce the condition that two point groups should be trans-
formable, the one into the other; viz. that all we should have
to do would be to extend the point groups, and then see
whether they were so transformable. We could not infer
from this, however, that the point groups would be transform-
able into one another by a point transformation, unless we
know that the contact transformation, which transforms the
one extended point group into the other extended point
group, is itself a mere extended point transformation.
§ 233. We have proved that given any system of structure
constants we can always find a contact group with the given
structure. The particular one we have shown how to construct
was an extended point group ; there will however be others ;
in fact, we have only to apply an arbitrary contact transfor-
mation to this extended point group, and we shall have a
group which will not generally be a mere extended point
group. Such contact groups, however, being deducible from
extended point groups by a contact transformation, are said to
be reducible contact transformation groups ; other groups
which have not this property are said to be irreducible.
The structure constants of any contact transformation
group, reducible or otherwise, satisfy the conditions
cikj + ckij = °>
t = n
2 (ciktcljs + ckjtctis + cjilctks) = °>
as we at once see from the identities
(f*,3)+ 0^.^=0,
((Wit Wh), Wj) + ((Wh, Wj), Wt) + ((Wj, W{), Wk) = 0.
§ 234. Contact transformation groups in z, x± , . . . , xn , p1 , . . . , pn
are point groups in these (2n + 1) variables ; but it is not true,
conversely, that point groups in (2n+l) variables are
necessarily, or generally, contact transformation groups. If
we write the variables in the form z,xx, ...,xn, p1, ...,pn, the
group in these variables will only be a contact transformation
one in the (u+l)-way space z,x1,...,xn if all the transfor-
mations of the group are characterized by the property of
leaving the equation
dz— pxdxx — ...— pndxn = 0
invariant.
From a knowledge then of contact transformation groups
234] PFAFF'S PROBLEM 293
in spaces of lower dimensions we can often deduce important
information as to point groups in space of higher dimensions.
Thus suppose, in space of s dimensions, we know that a group,
which we wish to determine, has the property of leaving
unaltered an equation of the form
f1dx1+...+fsdxs = 0,
where /1? ...,/s are functions of x1,...,xs. By the theory
of PfafFs Problem a transformation of the variables will
reduce this equation to one or other of the two forms
dym+i-Pidyi- -~-Pmdym = °>
p1dy1+...+pmdym = 0,
where 2 m + 1 does not exceed s ; and therefore the group
we seek must, when expressed in terms of the new variables,
be a contact transformation group in a space of not more than
\ (s + 1 ) dimensions.
CHAPTER XIX
THE EXTENDED INFINITESIMAL CONTACT
TRANSFORMATIONS : APPLICATIONS
TO GEOMETRY
§ 235. If z = (j> («!,..., xn) is any surface in (?i+l)-way
space, we shall now consider how the derivatives of z with
respect to oc1, ..., xn are transformed by the application of an
assigned infinitesimal contact transformation.
We must regard the function 4> which defines the surface as
unknown ; for otherwise the derivatives of z would be known
functions of xx , . . . , xn ; and the contact transformation would
be (when we replace pv ...,pn by their expressions in terms
of xv ..., xn obtained from z = ${xx, ..., xn)) a mere point
transformation ; and would apply, not to any surface, but
merely to the particular surface under consideration.
Let Pi,:.,pn be the first derivatives,^-,... the second
~b2Z
derivatives, where #,-,• denotes -r — ^ — > pi!k, ... the third
derivatives and so on ; and let W be the characteristic func-
tion of the assigned contact transformation which it is our
object to extend to derivatives of any required order.
Let the extended contact transformation be denoted by
z =■ z + 1 Q (xx , . . . , xn , 0, px 5 • • • j Pn) '
X^ = Xfht qi \XX , . . . , Xn , Z, 2^i j • • . j Pnfi
Pi — Pi + " ""« v^i ) • • • s xn ' ^"> Pi » • • •' Pn)>
Pij = Pij + ^7ry \xl* •*•' xn> z,> Pi' •••' Pn> Pll> '*•' Pin* P2I' •••)>
and so on, where in ir#, ... no derivatives of order higher
than the second can occur, in ir«j , . . . no derivatives of order
higher than the third, and so generally.
We know how to express (, £<s vit in terms of W and its
derivatives, and we have now to express similarly w#, . . . .
235] EXTENDED CONTACT OPERATOR 295
i = n
We have dp'k — 2 Pu dxi >
i — n i = n
and therefore d irk = 2 ^u dxi + *2,Pkid£i>
so that
i — n j = i = ?i i = n
(!) 2 *Mdxi-^Pkij£idxi = d(*k-^Pki£i)-
d
If we use the symbol -5 — to denote differentiation with
respect to xk, keeping sc1,...,Xje_1, xk+1, ..., xn all constant,
but not z or its derivatives, we have
Now -**=5^+**aJ* 6* 5S1
so that «i-ift,6= ~^-^^-^^^ = -^'
since W does not contain derivatives of order higher than
the first.
From the equation (1) we can therefore deduce
J? t d%W
I'M ZPhjitj- dx.dXh'
The result at which we have arrived may be thus stated :
dW
— ttj= — — s with the highest derivatives which occur omitted :
* dxi &
d2W
dxi dxk
omitted.
In exactly the same manner we could prove that
dzW
— Tr..h= ■ — j with the highest derivatives omitted,
VK dxidx,dxk
and so generally up to any assigned order ; and we thus see
how the infinitesimal contact transformation may be extended
as far as we please.
The extended contact operator is
i=u ^ >. i = n .. i-j=n ~
2^+^+2 ^+2^^: + ••••
— 77.,: , with the highest derivatives which occur
296
THE COEFFICIENTS OF THE
[235
If we have a group of infinitesimal contact operators then
these operators, when extended, will also form a group, of the
same order as the original group, and with the same set of
structure constants. This may be proved as in § 230, where
a like theorem was proved for the point group extended, so as
to be a contact group.
§ 236. It is convenient to have in explicit form the value
of the first few coefficients in the operators for the case n = 1
and n = 2, as they are required for applications to geometry
of two and three dimensions.
When n = 1, we take
and denote as usual
dy
dx
by P,
d2y
dx2
^ ?>
dxz
by r;
for - — hjOr-we shall write X, and we now have
dx dy
dW dW m
Also if
we have — k =
q'—q + tK, and r'=r + tp,
d2 W
-j-g- j with the highest derivative omitted,
= (X+ll)(X+!l)V:
dp>
dp-
and therefore, since q r— X— qX — = q — 3
dp dp z dy
Similarly
dzW
dxz
dp ' * dp2 dy'
~P
with the highest derivative omitted,
-(*+95;+'£)(*+*s+'s)(*+*£)»i
which, since
dp dq
d
dp dq
dp'
dp dp dy
237] OPERATOR IN EXPLICIT FORM 297
may be written
+,(iJ5?+,Xj_ + _)F.
§ 237. As an example of the application of these formulae
we shall find the form of those infinitesimal contact trans-
formations which transform straight lines of the plane into
straight lines.
The differential equation satisfied by all straight lines on
the plane is q = 0 ; and therefore, since we must have q'= 0,
we must have k = 0, wherever q = 0. We therefore have
X2W= 0 ; or, explicitly
WW n #W 2^W n
of which the general integral is
W = / (y-px, p) + X(j> (y -px, p).
Any contact transformation, whose characteristic has this
form, will transform any straight line into a straight line ;
these transformations have therefore the group property, but
the group is not a finite one.
If W1 and W2 are two characteristic functions of this group
the characteristic of the alternant of the operators Wr and W2
has, we know, the form {Wv W2) where
and (Wv W,) = XWX 3-IF2 -^ •
We know then that Wx and W2 being any functional forms
which satisfy the equation (1), {Wlt W2} will also be a func-
tional form satisfying the same equation. This result may
easily be verified independently.
If we only require those contact transformations which are
mere extended point transformations, then by (1), since
W=p£-r),
and £ and -q do not now involve p,
P in ~ Vn + 2P (P £12 ~Vn)+P2(P £22^22) = 0,
298 POINT GROUP TRANSFORMING [237
where the suffix 1 denotes differentiation with respect to x,
and the suffix 2 differentiation with respect to y.
Equating to zero the coefficients of the several powers of p
in this equation, we get
£>2= °> 111 = °» 7?22~2£l2 = °> £n-2T7i2 = °-
Differentiating these equations with respect to x and y, we
see that all derivatives of the third order are zero ; we
therefore take
£ = axx2 + 2kxxy + bxy2 + 2gxx + 2fxy + cv
77 = a2x2 + 2h2xy + b2y2 + 2g2x + 2f2y + c2.
From £22 = 7ju = 0
we conclude that a2 = bx = 0 ;
and from rj22 — 2 f12 = 0
we see that 2lix—b^ = 0 ;
while from £n — 2 tj12 = 0
we get 2 h2 = ax ; and we thus obtain
W = ax (2ix2—xy) + b2 (pxy—y2) + 2gxpx+ 2fxpy
+ cxp-2g2x-2f2y-c2.
W is therefore merely the most general characteristic function
of the extended projective group of the plane.
§ 238. We shall now find the form of those infinitesimal
point transformations which have the property of transforming
the circles of the plane into circles on the same plane.
The differential equation satisfied by all circles is
3q2p-(l+2)2)r = 0,
and we must therefore have
(1) (1+p2) p + 2prir-§pqK — 3q2ir = 0
for all values of x, y, p, q, r such that 3q2p = (1 +p2) r.
Since W=p£— rj,
and the contact transformation is now a mere extended point
transformation, W will only contain p in the first degree.
Applying the formulae of the preceding article to the equa-
tion (1), and substituting for r its equivalent expression in
terms of p>> q, we must have the equation
238] CIRCLES INTO CIRCLES 299
(2> +S12P(3X^ + ^)W
= (3?2 - rrj) zr + 6pq (X2+ 2?x4 + ? & w
satisfied for all values of x, y, p, q.
Equating the coefficients of q2 on each side of this equation
we have
Substituting for W the expression p$—r], where £ and 77 do
not contain p, this is equivalent to
= p(l+p2)(€1 + 2p£2-rl2).
Equating the coefficients of the different powers of p on each
side we get the two equations
(3) £i-i72 = °» & + *h = °-
Equating to zero the term in (2) which is independent of q,
we get XZW = 0 ; that is,
Pini + 3P2£ii2 + 3.P3fi22 +^£222 = ^111 + ZPVno + 32>2j?i22 +2^222;
and therefore, since p, x, y are unconnected,
Vvi = °> 3 ^112 — fin = °> *?i22 — £112 = °> ^222 — 3 f 122 = °> £222 = °-
If we differentiate the equations (3) twice with respect to x
and y, we shall see that all derivatives of £ and ri of the third
order must be zero.
We therefore take
£ = a1x2 + 2h1xy + b1y2 + 2g1x+2f1y + c1,
77 = a2x2 + 2h2xy + b2\f + 2g2x + 2f2y + c2 ;
and from the equations (3) we deduce that
ax = h2, ht = b2, gx =/2, a2 + hx = 0, bx + h2 = 0, g2+fx = 0,
so that the characteristic function is of the form
ax (p (x2 -y2)-2 xy) + a., (y2 -x2- 2pxy) + 2gx (px - y)
+ 2f1(inj + x) + c1p-c2.
300 CONTACT GROUP TRANSFORMING [238
It may at once be verified that for this value of W the co-
efficient of q vanishes in (2) ; and we thus see that there is
a point group of order six which transforms circles into
circles ; the six independent operators of the group are
d 3 a a o d
--, — , x- y^, x— +y — ,
ox oy oy ° ox ox oy
(x2—y2) r — \- Ixy — > 2xy- — I- (y2 — x2) — •
J 'ox J oy J ox J ' oy
Of these infinitesimal operators the first corresponds geo-
metrically to a small displacement along the axis of x ; the
second to a displacement along the axis of y ; the third to
a rotation round the origin ; the fourth to a uniform ex-
pansion from the origin ; the fifth to an inversion with
respect to a circle of unit radius whose centre is the origin,
succeeded by an inversion with respect to a circle of unit
radius whose centre is at x = t, where t is small, and lastly,
by a translation backwards along the axis of x measured by
t ; the sixth operator has a like interpretation with regard to
the axis of y. It is of course obvious that each of these
operations changes circles into neighbouring circles ; and we
have now proved that any infinitesimal transformation, which
does so, must be compounded of these six operations.
§ 239. We next try whether there are any infinitesimal
contact transformations — not mere extended point trans-
formations— which have this property.
If we substitute in
(1 +p2)p + 22ir-7T — SpqK— 3q2ir = 0
for p, k, it their values obtained in § 236 ; and then for r the
expression . ., » the resulting equation must be satisfied for
all values of x, y, j), q. Equating as before the coefficients of
the different powers of q to zero, we obtain
33 W o2W
(l+^(X_+_)
.W oW,
+ W-l)XW-W+l)(pX^ +p ~j =
239] CIRCLES INTO CIRCLES 301
X3W = 0.
From the first of these equations we see that
where A is a function of x and y only ; and therefore
and the second equation gives us a mere identity satisfied
i whatever function A may be.
The third equation gives
W=AVl+p2+Bp + C,
where A, B, G are functions not containing^?.
If this value of W is to satisfy the other equations it is clear
from the irrationality of V\ +p2 that AVl +p2 and Bp + C
must separately satisfy the equations. Now the latter part
would give rise to a mere extended point transformation ; and,
as we have fully discussed all the point transformations which
transform circles into circles, we need not further consider
this part, but have only to find what, if any, are the possible
values of the unknown function A.
Taking then W= A V 1 +p2,
*W Ap
we have
*P Vl +fi
2
(An + 2A12p + A22p*)pSl+p2 + (A12 + A22p)(l+p'i)*
= 2pV\+p2{An+2pA12+p>2Ai2),
which on dividing by Vl +£>" and equating the powers of
p gives
(1) An = A22) A12 = 0.
Finally the fourth equation gives
Aul+3AU2p + 3A122p2 + A22,pz= 0,
from which we see that all derivatives of A above the second
vanish ; and therefore
A = ax2 + 2hxy + by2 + 2gx + 2fy + c.
30.2 CONTACT GROUP REGARDED [239
From (1) we further see that h = 0, and a = 6, so that
A is the power of a circle.
The most general contact transformation group which trans-
forms circles into circles has therefore the following ten
characteristic functions :
(y2 + x2)Vl+2A yVl+2A xVl+2>\ Vl+p\
' ' p(x2 — y2) — 2xy, y2—x2 — 2pxy, px—y, py + x, p, 1.
§ 240. If we look on x, y, p as the coordinates of a point in
three-dimensional space, to a point there will correspond an
element of the plane ; and to two united elements of the
plane, that is, two consecutive elements whose coordinates
satisfy the equation
dy—pdx = 0,
there will correspond two consecutive points in space con-
nected by the equation
dy—pdx = 0.
If we write z for p we may say that to every transforma-
tion in space which leaves dy — zdx = 0 unaltered there
corresponds a contact transformation in the plane, and
conversely.
The group of contact transformations which we have just
found leaves unaltered the system of circles
x2 + y2 + 2gx + 2fy + c = 0,
and therefore also
x + g + (y+f)p = 0.
The corresponding group of point transformations in three-
dimensional space must therefore leave unaltered the system
of curves given by
x2 + y2+ 2gx+2fy + c = 0, x + g + (y +/) 0 = 0;
that is, will transform any curve of this system into some
other curve of the same system.
It is now convenient to write the equations of this family
of curves in the form
. . 4 c (x2 + y") + 4 (b2 — ac) (y + ix) +y — ix — a = 0,
^' Sc(x + yz) + i (b2 — ac)(z + c)+z-i = 0,
where a, b, c are variable parameters, and i is the symbol
for /3I.
240] AS A POINT GROUP 303
If we apply the transformation
(2) x'=y + ix, y'=y-ix, 2/ = ^J
which leaves unaltered the equation
dy—zdx = 0,
the equations (1) are transformed to
.. 4exy + 4:(b2 — ac)x + y—a = 0,
'' 4c(y + xz) + 4(b2 — ac) + z = 0 ;
so that the group into which the group (2) of § 239 is trans-
formed by the equations (2) of the present article leaves the
equations (3) unaltered.
Transform again with
x'
y = y'-\x'z\ x = -\^, z = -z'2,
which gives dy—zdx = dy'—z'dx' ;
and the equations (3) become transformed into
— 2cxy + cx2z— 2 (hr — ac) x + yz — \xz2 — az = 0,
4 cy + 4 (b2 — ac) —z2 = 0.
Eliminating z between these two equations we get
(ex2 — y + a)2 = 4 6- a2 ;
and therefore, since b is a variable parameter, we may write
these equations in the form
(4) y = c x2 + 2 bx + a, z = 2 b + 2 ex.
The group into which the group (2) of § 239 is now trans-
formed leaves the system (4) unaltered ; or, expressed as
a contact group in the plane, leaves invariant the system of
parabolas whose axes are parallel to the fixed line x = 0 ;
or, again, leaves unaltered the differential equation
The group into which (2) of § 239 is transformed could
have been directly obtained from this property of leaving (5)
unaltered, just as (though more simply than) the group which
left circles unaltered was obtained. If the group is thus
directly obtained, it will serve as an example of the applica-
304 A PROJECTIVE GROUP, ISOMORPHIC [240
tion of § 231, to prove that the two groups are transformable,
the one into the other, by a contact transformation.
§ 241. Let us next apply the point transformation in three-
dimensional space
x' = x, y' = y — \xz, z' ' — \z,
for which dy — zdx = dy' — z'dx' + x'dz',
and for which therefore a linear element of any curve in the
plane is transformed into a linear element of the linear
complex m = /3.
We then see that the group of contact transformations,
which leaves unaltered the system of parabolas, is transformed
into a group of point transformations in three-dimensional
space, with the property of leaving unaltered the system of
straight lines 7 7
° y = bx + a, z = ex + o ;
that is, into a projective group which does not alter the linear
complex m = /3.
We have thus established a correspondence between the
circles of a plane, and the straight lines of a linear complex
in space of three dimensions ; and the two groups, one a con-
tact transformation group in x, y, p, leaving the system of
circles unaltered, and the other a point group which trans-
forms the straight lines of a given linear complex inter se,
are transformable, the one into the other, by a point trans-
formation in three-dimensional space. It should be noticed,
however, that this point transformation is not a contact
transformation in x, y, p, such as was that which transformed
the system of circles into a system of parabolas.
If we write the equation of a circle in the plane in the form
(£-a)2 + (2/-/3)2 + y2=0,
then the group of transformations, which transform any one
circle into any other, being a contact group, will transform
two circles which touch into two other circles which touch.
Now we have seen, in Chapter VIII, that if a group trans-
forms an equation of the form
J (X^, .,., Xn, ttrj, .,,, drj = V
into another equation of like form, but with a different set of
parameters, then we can construct a group of transformations
in the variables ax, ..., ar, such that if X1,..., Xm are the
operators of the group in the letters xv .,., xn and Av ..., Am
242] WITH THE CONFORMAL GROUP 305
the operators in the letters alt ..., ar, the structure constants
of the two will be the same ; and each of the operators
will be admitted by the equation
If we apply this method to the system of circles on the
plane which admit the group (2) of § 239, we shall have
a group in the variables a, /8, y ; this group will be of the
tenth order, and will be found to be the group of conformal
transformations in three-dimensional space.
This result is obtained directly by Lie from the considera-
tion that the condition for two neighbouring circles touching is
da2 + d/32 + dy2 = 0;
for, since the transformed neighbouring circles must also
touch, the equation
da2 + d(32 + dy2 = 0
must be unaltered ; that is, the group must be the con-
formal one.
§ 242. We shall now write down in explicit form (for the
case n = 2) the values of the functions ttu, Tin, tt22 which in
future we shall denote by p, a-, r.
We have (p, q, r, s, t having their usual meaning)
W=pi + qv — C,
and the infinitesimal operator is
d z fc a a a a a i
^ <>x ty °z <>p <>q <>r °s °t
We denote by X and Y the respective operators
and we have
— + p tt- and r + gr-;
*F ^F _w SF , ^F w
CAMPBELL
306 THE EXTENDED CONTACT OPERATOR [242
d2W
da?
d2W
Since - p = -j-g- > with the highest derivatives omitted,
a a n / „ ^ d
=(x+^+s^)(x+^+s^)F-
and since ^— X— X— = ^— = r— F— F^— >
0/? <)/> 4)2; dq dq
we have
*\ "\ *\2 "\2 *\2 "\
-P = (X1 + 2 rX — + 2sX— + r2 ^ + 2rs ;— - + s2 — j + r — ) jr.
Similarly — 0- and — t are obtained from the operators
XF+sX^ + iX^-+rF^-+sF^- + r8^ + (^ + S2)^- + si~+,
d£> dq dp oq op* ' dp dq dq*
d d d2 d2 d2 d
and Y2 + 2sY^-+2tY—+s2^- + 2st^-^- + t2—2+t^.
dp dq dpr dp dq dq- dz
§ 243. As an example of the application of these formulae
we shall find the form of the most general infinitesimal
contact transformation which does not alter
d2z
= 0.
dxdy
Since we must have 0- = 0 wherever 8=0, we get
d2W n -TiW ^rdW ^Tfr A
= 0, X— - = 0, F-r— = 0, XFTF= 0.
dp dq dq dp
dW
From the first of these equations we see that - — does not
contain p ; and therefore by the second we must have
d2W d2W
dq dz dq dx
,dW.
so that — is a function of y and q only. Similarly we see
dq
that — is a function of x and p only, and therefore the
characteristic function W is of the form
f(p,x) + (f)(q,y) + ^(x,y,z).
243] IN THREE-DIMENSIONAL SPACE 307
Since XYW vanishes identically,
and therefore
^12 = °> ^23 = °> ^13 = °> ^33 = °>
so that i//- (x, y, z) = az + F(x) + 4> (2/),
where a is a mere constant and F and 4> functional forms.
The characteristic function which leaves unaltered the
equation s = 0 is therefore of the form
f(p,x) + <j>(q,y)+az.
There are therefore three distinct forms of characteristic
functions leaving s = 0 unaltered ; and, corresponding to these,
three distinct groups of contact transformations with this
property. Firstly, the infinite group where W is of the form
f(p, a), /being an arbitrary functional symbol ; the functions
of this group form a function system of the second order.
Secondly, the infinite group with characteristic functions of
the form <p (q, y), where <j> is an arbitrary functional symbol ; the
functions of this system also form a function system of
the second order, any function of which is in involution with
any function of the first system. Thirdly, the group with the
single characteristic function z ; if we form the alternant of
this function with any function of the first system, we have
another function of the first system ; and a similar result
follows for the alternant of z with any function of the second
system.
The infinite group of contact transformations leaving un-
altered the equation s = 0 is compounded of the operations
of these three groups.
We have proved that any Amperian equation with inter-
mediary integrals of the form
ui = /1 (^i) and u2 =/a (v2)>
where /x and/2 are arbitrary functional forms, can by a con-
tact transformation be reduced to the form s = 0.
It follows that any such Amperian equation will admit an
infinite group of infinitesimal contact transformations, the
operators of which may be arranged in classes as follows : in
the first class there are two unconnected operators, but an
infinite number of independent operators : in the second class
there are also two unconnected operators, and an infinite
number of independent operators : in the third class there
is only one operator: any operator of the first class is
X 2
308 TRANSFORMATIONS WHICH DO NOT [243
permutable with any of the second, and the alternant of the
operator of the third class with any operator of one of the
other classes is an operator of that other class.
§ 244. We have obtained the conformal group in three-
dimensional space from the property that it leaves the equation
dx2 + dy2 + dz2 = 0
unaltered ; if we seek the group which will leave the expression
dx2 + dy2 + dz2
unaltered, we shall obtain the group of movements of a rigid
body.
The question now proposed is to find the infinitesimal point
transformations which have the property of transforming a
given surface into a neighbouring one, without altering the
length of arcs on the surface ; that is, if P and Q are any
two neighbouring points on a given surface which receive
infinitesimal displacements so as to become two near points
P\ Q' on a neighbouring surface, we want to find the relations
between £, rj, £ in order that we may have PQ = P' Q'.
Since we must have
dxd£+dydrj + dzdC= 0
for all values of x, y, z on the given surface ; and
di = fxdx + %**
7 „ d ( 7 d C 7
dC=didx+Tydy<
dr)
di
, arj 7 a?? 7
d"=d-xdx+Tydy'
dz = pdx + qdy,
we get, by equating the coefficients of dx2, dxdy, dy2 to zero,
d£ , „d(
& ~fx+Vfx="°>
d£ drj d( d(
dy dx dy ^ dx ~
dr] d( _
dy dy~" '
where -r- and -j- denote total differentiation with respect to x
dx dy
and to y.
From the equations
d2 ,d£
drj
dx
dC
dy
9f^) = °>
dx-
dxdy v dy
d£ d{s a" ,ar\ au
dcr. " Use.) ' dx'^dy dy '
dy2 \dx dx
244] ALTER THE LENGTHS OF ARCS 309
we can eliminate £ and g, and thus obtain the equation
(2) td*^ 2s d*£ r^-0
dx2 dx dy dy2
The surface on which P and Q lie is a known one, and
therefore r, s, t are known in terms of x, y, so that the equa-
tion (2) determines £ as a function of x and y.
From
d ,d£ dr] d( d£\ d ,dr\ d£\
dy^dy dx dy dx' ' dx^dy ^ dy' "
e et d?£ dH tdC_Q.
° dy'1 dy1 dx~~ '
while by differentiating
dx dx"
with respect to # and with respect to y we get
^| ^c r^"_0 and _^!l_ _^!£. ^r_
cte2 cfoc2 cisc ' dxdy dxdy dx ~
with similar equations for rj.
If we denote -^ ^ by X
dx dy
we have, therefore,
^ dxdy y <&c2 dx dy-'
( ^!i_ ^ t d*£ ^W
^ dy'2 dxdy dxdy dy'
which is a perfect differential, since
dx2 dxdy dx2 '
and therefore A can be obtained by quadratures, when f is
known in terms of x, y.
When we know A. and £ the derivatives of £ and 77 are
known by (1) ; and therefore £ and 77 can be obtained by
quadratures. It will also be noticed that when ( is fixed,
£ and 77 are fixed, save as to the terms ay + b in £ and —ax + c
310 TRANSFORMATIONS WHICH DO NOT [244
in 7] where a, b, c are arbitrary constants. The infinitesimal
transformation is therefore fixed when ( is fixed, except for
small translations along the axes of x and y, and rotations
round the axis of z.
The mistake of supposing that the operators
ox oy dz
so found will generate a group must be guarded against : if U
is an operation which transforms a surface 8 into 2 and pre-
serves unaltered the lengths of small arcs on S, and V is
another operation with the same property, then VU will not
necessarily have the required property, because V has not
necessarily such a property for the surface 2.
§ 245. We can now employ the values of p, a, t obtained in
§ 242 to prove the known theorem, that any such infinitesimal
transformation as we are now considering will so transform $
into 2, that the measure of curvature will be the same at
corresponding points on these surfaces.
We have
-tp-rT + 2s<r = (t -~+r-rii-28-r-r)W,
v ax* dy ax ay''
omitting derivatives of the highest order which occur, that is,
derivatives of the third order ; and this expression is equal to
d2 d2 d2
(f -\- T 2s )(v^-\-oti)
^ dx2 dy2 dxdy' ^ '
since (t -j-2+ r -j-„ — 2s -j— r) ( = 0.
v ax* dy2 dxdy'
Now
(t^L. r— 28-^-} £- (t— r d* 2s d* V
^ dx2 dy2 dxdy' ^ dx2 dy2 dxdy'
. d£ d£ „ cZ£ 0d£
+ 2tr-r± + 2sr-^ — 2sr ~ - 2s2 -^ ,
dx dy dy dx
the other terms being omitted as they are derivatives of the
third order.
If we now make use of the equations (1) of § 244 to express
the derivatives of £ of the second order in terms of those of (,
we have
246] ALTER THE LENGTHS OF ARCS 311
by (1) of § 244.
Similarly we see that
/, d2 d2 d2 \ dC
Again
d d
-PTT-qK = fp -j-+q -=-) {p£+q-n-Q, omitting the highest
iXJj Lit/ j • ■•
•^ derivatives
drc ^2/ ^ <% dx ' dx~^dy
= _ (1 +p2 + g*) (p g + ? ^) by (1) of § 244.
Now in order to prove that the measure of curvature is
unaltered by the given infinitesimal transformations, it is only
necessary to prove that
(l+p2 + q2)(tP + rT— 2sa) = ±(rt — s-)(pir + qK);
and this is at once proved by aid of the formulae now
obtained.
§ 246. If we have an go2 of points on a surface and the
distance between neighbouring points (measured along a geo-
desic on the surface) is invariable as this go2 of points moves
on the surface, we then have on the surface the analogue of
a rigid lamina in a plane. Such an assemblage we call a net ;
and the question is suggested, can a movable net exist on any
surface, or can it only exist on particular classes of surfaces 1
If P is any point on the net which moves to a neighbouring
point P', we have just proved that the measure of curvature
at P and Pf must be the same ; we shall first discuss the case
where the given surface has not everywhere the same measure
of curvature.
Through each point on the surface draw the curve along
which the measure of curvature is constant, and let these
312 SURFACES OVER WHICH [246
curves be called the curves of constant curvature. Next draw
the system of curves cutting these curves of constant curvature
orthogonally, and call these latter curves the trajectories.
Let Ax, A2, ... be a series of neighbouring points on a tra-
jectory ; if the set is movable A1,A2,... will take up positions
Br, B„,... and the points of the net which were at BX,B2, ...
originally will now take up a position Cls C2,... and so on.
The points A1,B1,C1,... must lie on a line of constant
curvature; similarly A2,B2,C2,... must lie on such a line,
A3, B3, (73, ... on another, and so on. It will now be proved
that this net movement is only possible if Bx, B2, ... lie on a
trajectory, C1? C2, ... also on a trajectory, and so on.
Since Ax Bx = Bx Gt and A1 A2 = Bx B2 and A2 Bx = B2Clt
it follows that the angle A2A1B1 = B2B1C1; and therefore,
since A2 A1 Bx is a right angle, so is B2 Bx Gx ; that is, Blt B2i ...
lie on a trajectory.
Unless then the surface is such that trajectories can be
drawn on it, dividing each line, along which the measure of
curvature is constant, into the same number of equal parts,
the surface cannot allow a net to move over it. If this con-
dition is satisfied, and the surface be not one with the same
measure of curvature everywhere, the net can move on it
with one, and only one, degree of freedom.
Since A1 A2 = Bx B2 the perpendicular distance between
two neighbouring lines of constant curvature is the same at
all points ; it therefore follows that the trajectories are geo-
desies on the surface.
If we take u and v to be the coordinates of any point on
the surface, where u = a and v = /3 are respectively the lines
of constant curvature and their trajectories, we can take for
the element of length on the surface
ds2 - du2 + \2dv*
when X is a function of u only.
If the net is to have two degrees of freedom in its move-
ments the surface must be everywhere of the same measure of
curvature.
§ 247. We can prove these results in a different manner and
also obtain all possible movements of the net, if we employ
surface coordinates.
Let the equation of the surface be given in the form
x = /j (u, v), y = f2 (u, v), z=f2 (u, v),
247] A NET CAN MOVE 313
so that we have
ds2 = edu2 + 2fdu dv + gdv2,
where e, f, g are functions of the parameters u, v which define
the position of any point on the surface.
We shall first prove that by proper choice of the parameters
we may take e = 1 , / = 0, and thus simplify the expression
for the element of length.
We must prove that we can find p and q, a pair of functions
of u and v such that
edu2 + 2fdudv + gdv2 = dp2 + X2dq2.
Since
dp = ~- du + 7T- dv and dq = ^-dw + ~ dv,
r cu vv ou ov
we at once obtain as the necessary and sufficient conditions for
such reduction
i>V' ^dv
'-$&+*&>:
and therefore
H^)>-(s?)')=t^)a-
It follows that p must satisfy the equation
When we have thus determined p as a function of u and v,
we can determine A and q by the equations
eliminating g we have, for determining A, the equation
2)
3
v /y Uu^ " t>u ^/" vdv'
When A is thus determined we can find q by quadratures.
We have therefore proved the theorem we stated, viz. that by
314 SURFACES OVER WHICH [247
a suitable choice of surface coordinates we may take
(1) da2 = dp2+tfdq2.
If we form the differential equation of the geodesies on the
surface with respect to this system of coordinates, we shall
see that it is satisfied by the curves q = constant: these
curves are therefore geodesies.
§ 248. We can throw this expression into another form
which will also be required in our investigation ; take a new
set of parameters such that
dp + iXdq = ixda and dp — iXdq = vdfi,
where i is the symbol for V — 1 ; that is, - is the integrating
1 **
factor of dp + iXdq and - the corresponding factor for
dp — iXdq-, we now have
ds2 = eh da dp,
where h is some function of a and /3.
It is convenient to write x for a and y for /3 so that
ds2 = ehdxdy.
Suppose now that points on the surface admit the in-
finitesimal transformation
x'=x + t£(x,y), y'=y + tr](x,y),
which does not alter the length of arcs ; that is, suppose that
a movable net can exist on the surface.
Since ds is to be unaltered we must have
dx drj + dy dg+ dxdy (£— + tj — ) h = 0 ;
v ox oy'
and therefore by equating the coefficients of dx2, dxdy, dy2
to zero we get
dx oy ox oy ^ox oy'
From these equations we conclude that £ is a function of
x only, and r\ a function of y only ; and therefore, by taking
as parameters, instead of x, a suitable function of x, and,
instead of y, a suitable function of y, we may in the new
coordinates take £ and ?? each to be unity. In fact if £ =f{x)
249] A NET CAN MOVE 315
then from x'= x + tf(x),
we conclude that whatever $ may be,
4>{rf) = 4>{x)+ij W(a);
if then we take <$>'(x)f(x) to be unity, and $(x) as a new
parameter in place of x, £ will be unity.
Since we must now have with these parameters
oh oh
ox cy
h must be a function of x— y.
We can therefore, if the surface can have a movable net
drawn on it, so choose our surface coordinates that
<!»+4(^i=i))2=0;
v ox J
where / is some functional symbol ; and we have
. o
ox
-\ 2
ds2 = - ( — f(x-y)) ((dx + dyf - (dx - dyf)
= WT-C/J(dx + dy)*.
ox
This form is the same as (1) of § 247, only that X2 is now
a function of p only and not of q ; and we conclude that the
net can move, if and only if, the element of arc can be written
in the form ds2 = dp2 + \2dq2
where A2 is a function of p only.
§ 249. We now assume the surface to be such that we may
take ds2 = dx2 + k2dy2
where A is a function of x only.
It is known (Salmon, Geometry of Three Dimensions, § 389)
d2k
that the measure of curvature is y-^ -r A ; and therefore the
lines on the surface where the measure of curvature is con-
stant are the lines x = constant.
' To find the most general displacement of the net on the
surface we now have
dxdg+\2dydrj + dy2(£~ +T]^t)>
316 SURFACES OVER WHICH [249
and therefore, since A does not contain y,
(1) »! = 0, (2) K"U + U- 0, (3) A* »S +(** = 0.
v ' <>x <)x 2>y / t>y hx
Eliminating -q from the second and third of these equations
we get
and therefore
From the first equation we see that £ is a function of y only.
First suppose that £ is zero, then
i)X dy
and we get the possible displacement
x = x, y'=y + t;
that is, a displacement along a line where the measure of
curvature is constant.
If £ is not zero, since
*2£ i- «A *** ^x>2n
iL5 t..o(x°-± (Z±Y\
and £ is a function of y, and A a function of #, each of these
equal expressions must be a mere constant.
Suppose that this constant is not zero, then
dxl ^dx*
Solving this equation we get
A = rcosh (kx + e),
where e and k are constants ; and this value of A gives the
measure of curvature constant everywhere on the surface, and
equal to k2.
From ^l-2a2i,
dyz b
we get £ = A cosh V2ay + B sinh V2ay ;
249] A NET CAN MOVE 317
and from (2) and (3) we now have
\/2& -
r) = tanh (kx + e) (A sinh V2ay + B cosh V2 ay) + C,
ct
where A, B, C are arbitrary constants.
If we take A -j—2 — (-7-) to be negative and equal to — a2,
we should take A to be t cos (kx + e), and
£ = A cos </ 2 ay + B sin. \/f2ay,
*/2k -
r\ = — — tan (kx + e) (A sin V2ay — B cos -J 2 ay) + G ;
the measure of curvature at any point of the surface is
then equal to — k2.
By properly choosing the initial line from which x is to be
measured we may take e to be - when A becomes — j sin kx.
£i tC
In particular when k is zero, that is, when the surface is
a developable,
X=—ax, £ = A cos V2ay + B sin V2ay,
r) — (A sin \f2ay — B cos V2ay) + C.
(LOO
In general, then, we have three linear operators corresponding
to the three possible infinitesimal displacements of the net ;
d2k dK 2
and for the case where X-j—2 — ("7") is negative and not zero
these operators are Xlt X2, X3 where
X, = cos a/2 ay z V2~ - cot kx sin V2 ay — j
1 * ^>x a a <>y
X = sin V2ay- — 1- V2 - cot kx cos </2ay c- »
2 J cȣc a ^ ^2/
We obtain by simple calculation
(X3, ZJ = - V^oZj,, (X2,_X) = - v^a^,
(X1SX2) = A3.
318 GENERAL CONCLUSIONS [249
The discussion of the case where A -=-3 — ( -j- ) is zero may
be left to the reader ; it need only be stated that it cannot be
deduced from the results given by merely taking a to be zero.
The general result of this discussion is therefore to show
that, if a surface is not one over which the measure of
curvature is everywhere the same, at the most there can be
but one degree of freedom in the motion of the net ; and also
that no movement of the net is possible at all, unless the
surface is such that the perpendicular distance between any
two neighbouring lines, along each of which the measure of
curvature is constant, is the same at all points of the line.
On surfaces, however, with a constant measure of curvature
the net can move with three degrees of freedom ; and the
movements of the net generate a group of the third order.
This group will contain a pair of permutable operators if the
surface is a developable.
CHAPTER XX
DIFFERENTIAL INVARIANTS
§ 250. If we are given any function of 0, xx, ...,xn we know
that there are n unconnected linear operators which will
annihilate the function ; these operators form a group, though
not necessarily a finite group, with respect to which the given
function is invariant: and more generally, if we are given
m such functions of the variables fv ...,/m, there will be
(n+1— m) unconnected operators forming a group, with
respect to which flt ...,fm will be invariants.
So too when we are given a linear partial differential
equation of the first order, or a complete system of such
equations, we have seen in Chapter VII how the system must
admit a complete system of linear operators generating a
group. If the system of equations is of the first order, but
not linear, then, though it will not generally admit any
group of point transformations, yet it will admit a group
of contact transformations. In particular cases the equations
when not linear may admit groups of point transformations ;
thus we found (§§ 33-35) that the equation
admitted the conformal group of three-dimensional space.
In general, differential equations of order above the first
do not admit point transformation groups, but some particular
equations do ; thus
■ — - = 0
dx2
admits the projective group of the plane ; the expression
V*(Sf\
dh/\2)f d2y
dxA
for the radius of curvature admits the group of movements of
a rigid lamina in the plane. If we are given any differential
320 INVARIANTS OF A KNOWN GROUP [250
expression or equation, we have seen in Chapter XIX how-
to determine the infinitesimal point transformations which
it may admit; we have also considered examples of deter-
mining the transformations admitted by equations of the form
J [Z, X-^, . . ,, Xn, G/X^ , . . . , CLXn) = U,
and we have seen how closely all these different problems
are connected with the idea of extended point groups. The
method common to the solution of these problems was that of
determining the group admitted by a given expression (or
equation) which expression is then an invariant of the group ;
that is, the invariant was given, and the group was then
to be found.
§ 251. In this chapter we shall consider the converse
problem, viz. how, when the group is given, we are to obtain
the functions of z, xx, ...,xn, and the derivatives of z, which
preserve their form under all the operations of the group;
in other words, we are to investigate how the differential
invariants of known groups are to be calculated. We
confine ourselves to the case where the group is a finite
continuous one.
We have solved a part of the proposed problem in Chapter
VIII, where we showed how to obtain the functions of
z,x1, ...,xn which are invariant for a known group, and also
how to find all the equations which the group admits. Such
functions, or equations, may be considered as respectively
differential invariants of zero order or differential equations
of zero order ; and we have seen that only intransitive groups
can have differential invariants of zero order, whilst im-
primitive groups must have an invariant system of differential
equations of the first order.
Suppose that we now wish to find all the differential
invariants of the kth order of a known group, that is, in-
variants involving derivatives of the kth order. We first
extend the operators of the group to the kth order, when
we shall have the operators of a group in the variables
z, j»j, ...,xn, and the derivatives of z up to the kth order ; this
group has the same structure constants as the given group
We then apply the general method to this extended group,
and find its differential invariants of zero order, and these
will be differential invariants of the original group involving
the kth derivatives of 0; that is, they will be what we have
called invariants of the kth order.
252] EXAMPLE 321
In exactly the same manner, we see how the problem
of finding the invariant differential equations of the kth order
of the given group is reduced to that of finding those of zero
order in a group where the variables are z,x1} ...,xn, and the
derivatives of z up to the kth order.
§ 252. Example. As a very simple example, let it be
required to find the differential invariants of the third order
for the group
, , ay + b
x = x, y = — — , •
J cy + d
The linear operators of this group are
a a a
<>y a <)y * <)y
Now 7? — extended to the third order is
d y
vDy W*y)*H ^ *y*+y*?>yhy2
where we denote the first three derivatives of y with respect
to x by y1,y2, Vz respectively.
If we let 7] successively take the values 1 , y, y2, we see that
the functions we require must be annihilated oy the three
operators
V y*y ^tyi ***% **%'
and therefore also by the three unconnected operators
It follows that any function of x and ■ xJz „ — ^- will be
2/i2
a differential invariant of the required class.
It may similarly be shown, by further extending the
operators, that a differential invariant of the fourth order will
CAMPBELL
322 THE EXTENDED OPERATORS [252
have the three annihilators
d } 7> d d
a 2/ ^2/1 Hi ^2/3 ty*
2/12 4 + 32/l2/24 +^4^ + 32/22) ^ ;
that is, it will not involve y, will be homogeneous and of zero
degree in 2/i>2/2'2/3>2/4' and wiU De annihilated by the operator,
So also the invariant of the fifth order will not involve 2/,
will be homogeneous, and of zero degree in ylt ...,ys, and
will have the annihilator
Vi^- +32/i2/2^ + (42/i2/3 + 32/22)^ +(5 2/i2/4 + 10^2/3)^;
and so on, the new coefiicient of the next highest partial
operator being derived from the last by differentiating it
totally with respect to x, and adding unity to the coefficient
°f 2/i 2/5 obtained by such differentiation.
§ 253. We shall now write down the extended operators of
the projective group of the plane
a a a a a
(3) x- — \-y c y9^ 2w,- 31/4^ ...;
w *x **y U2Mj2 JdMj3 J*Mj±
a a a a j)
(4) #- V- 2^^ 3u7 c 4?/.- ... ;
w 7>x J*y J1MJx J^y2 j3Dy3
a a , , a a
d# ^2/ tyl ^2/2
-(42/i2/3 + 32/22)^3----J
the coefficient of — r — being obtained from that of — rr —
^2/4 ^2/3
by differentiating the latter totally with respect to x, and
adding unity to the coefficient of y1yi in the result, and so on ;
<6> y^x+x^-w-1hl1-3yly2ii~^■■'
253] OF THE PROJECTIVE GROUP 323
all terms after the third being the same as in (5) ;
-(5xy3+3y2) — -...,
the coefficient of — r — being formed by adding x2yr to the
coefficient of — r ■> differentiating the result totally with
respect to x, and omitting the highest derivative in the result ;
~(4 xyxyz + 3xy? + 32/i2/2 + 2/2/3) ty -••"
the coefficients of the successive terms being derived from the
preceding ones as in (7), only that instead of adding x2yr we
add xyyr.
We could now find the invariant differential equations and
the differential invariants up to any assigned order of this
group, or of any of its sub-groups. Thus (1) and (2) form
a sub-group of which any function of the derivatives not
containing x or y is an invariant; (1), (2), (3), (4) form a
sub-group of which any function of the derivatives y1}y2, ...
which is of zero degree and of zero weight will be an invariant ;
(1), (2), (5) is the group of movements in the plane with the
geometrically obvious invariants p, -~> -p-^ •••> where p is the
expression for the radius of curvature in Cartesian coordinates.
In order to obtain the differential invariants of a less
obvious group we take (1), (2), (3), (4), and (7) which is at
once seen to generate a sub-group. A differential invariant
of this sub-group must be a function oi yx,y2, ... of zero
degree and of zero weight ; the only other condition which
this function has to satisfy is that of being annihilated by
a z a d
(9) 3u, — - +82/0 — + 15y.<— + ... +r (r— 2)w_ , ,— + ....
It can be at once verified that the operator (9) annihilates
y a
324 A DIFFERENTIAL INVARIANT [253
h = 3»4-4#32. J5= 92/222/5 + 40 y.3-45 2/22/32/4'
I, = 3 V/Vg ~ 24 2//2/a2/r, + 60 y.Ajiy- 40 y:J4 ,
J7 = 27 2/24277-315 2/232/;,2/G+12G0 2//2/:i2zy5-2100 2/222/322/4
+ H202/35.
Each of the equations 74 = 0, I. = 0, ... is invariant under
the operations of this sub-group ; and one of these, I5 = 0, is
invariant under all the operations of the general projective
group of the plane. This last result is obvious from the
geometrical fact that I5 = 0 is the differential equation of the
conic given by the general equation of the second degree in
Cartesian coordinates. That y2 = 0 is an invariant equation
of the general projective group is also obvious geometrically.
The differential invariants of the sub-group (1), (2), (3), (4),
(7), as distinguished, from the invariant differential equations
of the sub-group, are up to the 7th order
V 76 I* y*IA
j 3 ' r 2 ' j 5 ' v
si J4 J4 2/:
4
What we have called invariant differential equations are
sometimes called differential invariants ; in such a notation
our differential invariants are called absolute differential
invariants.
§ 254. We now wish to find the differential invariant of
lowest order of the general projective group of the plane.
We anticipate ' by counting the constants ' that it will be
of the 7th order ; for there are eight operators in the group,
and we do not therefore expect an invariant till these operators
are extended so as to be in nine variables, and thus the
derivatives of the 7th order will be involved. We shall find
that this anticipation will be verified.
From (1) and (2) of § 253 we see that the invariant cannot
contain x or y ; and from (5) and (6) of the same article
we know that it will not contain y1 ; it must therefore be
a function of
I2 I I2
5 6 nnrl 7
TV Tl and TV
J4 J4 J4
since an invariant of the group must clearly be an invariant
of any sub-group, and therefore of the sub-group (1), (2), (3),
(4), (7).
If we now extend all the operators to the 7th order we shall
find that there are two additional operators to be added to
254] OF THE PROJECTIVE GROUP 325
(3), (4), and (9) of § 253 ; and that the invariant, which is
a function of y2, ...,y7 of zero degree and of zero weight,
must be annihilated by these operators. These new operators
are, omitting the parts of these operators which are connected
with (3), (4), and (9), (we may do this since these parts will
necessarily annihilate the invariant),
(10) 6y«2^ + BOfcy, s— + (80y,y4+40y,«) ~
+ (1052/22/5+ 1752/32/4> ^r '
and (11) 2y2y^— + l0y32^- + (35y3yi-3y2y5)~
+ (562/32/5 + 352/42-72/22/6)A.
The linear operator i)i
u T
(where (~r-^) denotes the total derivative of 75 with respect
to x) is connected with (10) and (11) ; and therefore we may
replace the annihilator (11) of the required invariant by the
annihilator (12).
Denoting the operators (10) and (12) respectively by X and
Y the invariant required is a function of
I2 T I2
J4 J4 J4
annihilated by X and F.
Now we easily verify that
XJ4=182/23, XJ6 = 0, XI, = BOy,"/,, XI, = 315^,
and therefore X annihilates P and Q, where
376-5742 2J7-35J4J5
and the invariant required will be that function of P and (^
which is annihilated by Y.
Now we may verify that
IT
and also that y2 ( -^) -5y.J5 = 3 i"6 - 5 14 .
326 SOME DIFFERENTIAL INVARIANTS [254
We then have
YQ=z1l^llp and F(7P2-Q) = 0;
and therefore 7 P2 — Q is the invariant which we require ;
that is,
(where 74, 75, 76, 77 are as defined in § 253) is the differential
invariant of lowest order for the general projective group of
the plane.
From this invariant we can deduce the differential equation
satisfied by all cuspidal cubics. To obtain this equation we
reduce the cubic by a projective transformation to the form
y2 = x6, and we therefore have
y = %%, 2/1 = #<#*> V-z = t»"*» 2/3 = -f ari
If we now calculate for this cubic the values of 74, 75, 76, 77,
and if we let 7 denote the numerator in (13), we have with
little labour a" 10«. P + 7» tf.I,* * 0;
and, as this equation is invariant for any projective trans-
formation, it is zero for a cuspidal cubic, given by any equation
in Cartesian coordinates.
§ 255. As an example in finding invariants of groups in
three-dimensional space, we might take the group of move-
ments of a rigid body, viz.
0000000 3 3
ox oy oz oz oy Ox Oz oy ° ox
and we should thus obtain the invariant differential equation
of the first order
/0Z\2 /OzJ*
and two differential invariants of the second order, viz. the
expressions for the sum and product of the two principal radii
of curvature at any point of a surface.
Since, however, these results are obvious geometrically we
shall consider instead the invariants of the group
3 3 o 3 _ 3 . , o o
^- + 2/— , *r- + 3r) x2^ + (xy—z)—+xz^->
ox J oz ox oz ox v J ' oy oz
255] IN THREE-DIMENSIONAL SPACE 327
a a d a . a . a a
:— + £c^-j v ^ — V z— i (xy—z) xr — h V n — H ^ ^— ;
Tiy ^z u Mj c>0 v * '<)# * Zy * lz
these are the operators of the group of movements of a rigid
body in non-Euclidean space.
Taking as usual p, q, r, s, t to denote the first and second
derivatives of z with respect to x and y, the twice extended
linear operator £- — \- v - — \- Ct-
r dx dy dz
.. d d .3 3 d 2> S 3
d£C d?/ 02; <>p ^>q ^>f os t>£
where (denoting by the suffixes 1, 2, 3 the partial derivatives
of £, 77, or £ with respect to x, y, z, respectively)
-^ = p2i,+pqvz+2H^i-Q + qvi-Ci,
-k = q2r]z+pq£2 + q{yi2-Q+p£2-t2,
-p =p3 ^+p2q^+p2{2^-Czz) + 2pqvu +p(£u-2Clz) + qrin
- Cn + 2 r (& +p £3) + 2 s (77! + ^773) + r (p£3 + qVz- Q,
-<r = p2q^ +pq%3 + p2^ + q2vn +Pq (£13 + %j + C33)
+P (£12 ~ C23) + q faia - &a) ~ C12
+ s(t1 + V2-(z + 2P& + 2q%) + r(£2 + qQ + t(7ll+2)r,z),
-r = q3Vzs + tfptn + q2 (2 ^23 ~ C33) + 2pq isa + q (V22 ~ 2 C23)
+p^2-C22+2t(ri2+qv-) + 2s(i2+qQ+t(p^+qv3-Cz)'
There are six sets of values of £, 77, £ viz.
(1) i=h v = o, C=y,
(2) g=X, V = 0, C=Z,
(3) £ = x2, 77 = xy-z, C— %z,
(4) £=0, r,= l, C=z,
(5) £=0, v=y, C=z,
(6) g = xy-z, v=y2, C=yz.
Forming by aid of the above formulae the corresponding
values of ir, k, p, a, r, we get the six operators
a d 2» 00
v ' c»oj 00 a dg or H
328 SOME DIFFERENTIAL INVARIANTS [255
(3) x^ + izy-^^ + xz^-frx + qy-z-pq)^
+ ^^-(r('Sx-q) + 2Hy-p))^
t*\ a 3 3
(4) — + X— + -;
cy dz 7>p '
,-v S a d , ^ ^
(5) **j+§*;+**p-iit+r&
(6) (xy-z)±+y*± + yz±-(px + qy-z-pq)±
+ P2^-r(U-^P)^-(^(y-p) + r(x^q))~
-(t(Sy-p) + 28(x-q))~,
§ 256. As we have six operators forming a complete system
in eight variables we expect two differential invariants of the
second order ; and could not have more, unless the six
operators are connected; and it is easily seen that they are
unconnected.
From (1) and (4) we see that the invariants must be
functions of p—y, q — x, z — ocy, r, s, and t ; we therefore write
p=P-y, Q = q-x, Z-z-xy,
The operator (2) now takes the form
and (5) the form
PTP+ZiZ+rTr'tYV
while (3) becomes
and we have a similar expression for (6).
256] IN THREE-DIMENSIONAL SPACE 329
It is now convenient to denote P by p, Q by q, and Z by z ;
in this notation we see that the invariants are functions of
p, q, z, r, s, t, annihilated by each of the four operators
d d d d
* * * .3
(y) (2 * +M) $z + f jz + (n + 28P)yr
+ (2sq + tp)*-s+3tq±,
(8) (2*+M)- + p2- +(^+28gr)l
+ (2sp + rg)-+3ri9— ,
which we denote respectively by I21} i22, 123, and I24.
We have
fii to + s) = M + z> ^2 to + z) ~ M + 0>
^3 to + «) = 2 2 (* +M ^4 to + z)=2p(z +pq),
so that the equation z +pq — 0 is invariant (or in the original
notation z + pq = px + qy).
Also £2j rq2 = rq2, i22 rq2 = rq2,
i23 rq2 = q2 (3 rq + 2 s_£>), £24 rg2 = (42;+ 5pg) rq ;
and forming similar equations for tp2 and s(j?g + 20) we see
that
ni(rq2 + t232-2s(pq + 2z)) = rq2 + tp2 — 2s(pq + 2z);
n2(rq2 + tp2-2s(pq + 2z)) = rq2 + tp2 -2s(pq + 2z) ;
0, (rg2 + tp2 -2s(2iq + 2z)) = 3q (rq2 + £p2 -2s(pq+2z));
&4 (rg2 + ^2 - 2 s (pg + 2 0)) = 3p (rq2 + tp2-2s(pq + 2 z)).
Since
f2i to + 0)^ s~* — to+z)^-* = ^2 to +2)*^'
123(£>g + 2)*2~"* = 3g(235' + »)^s_^,
330 GEOMETRICAL CONSIDERATIONS [256
we can therefore see that
rq2 + tp2 — 2 s (pq + 2 z)
(rpq + z)^z~^
is a differential invariant for the group.
It may be similarly proved that
rt—s2
(pq + z)2z~2
is the other differential invariant of the group.
In the original notation, therefore, the invariants are
r (q — x)2 + t (p — y)2 — 2s (2z+pq—px — qy — xy)
and
{z +pq —px — qy)% (z — xy)~*
rt-s2
(z +pq —px — qyf (z — xy)~2
§ 257. These examples indicate that the only difficulty in
obtaining differential invariants of a given group is the
difficulty of finding the solutions of a given complete system
of equations.
We are often much helped by geometrical considerations ;
thus in the example just considered we knew that the group
was a projective one in ordinary three-dimensional space ;
and we knew that it transformed the quadric z = xy into
itself. If then from any point P on a surface S we draw the
tangent cone to this quadric it will meet the tangent plane at
P to the surface S in a pair of lines ; these lines, together with
the inflexional tangents to S at P, will form a pencil of four
rays. The condition that the pencil should be harmonic is
unaltered by any projective transformation, and is, in the
notation here employed,
r(q — x)2 + t(p—y)2—2s(2z+pq—px — qy — xy) = 0.
Similarly the condition that the surface S should be a
developable is unaltered by projective transformation, and
is rt — s2 = 0.
It was by attending to these considerations that one was
enabled to simplify the solution of the given complete system.
CHAPTER XXI
THE GROUPS OF THE STRAIGHT LINE, AND THE
PRIMITIVE GROUPS OF THE PLANE
§ 258. When we are given the structure constants of a
group we have seen how the types of groups with the required
structure are to be formed. If, instead of being given the
structure constants, we are merely given the order r of the
group required, we should have to find the sets of r3 constants
which will satisfy the equations
h = r
Cijk + Cjik = °>
i (cahcjhm + ckjhcihm + cjihckhm) — °>
where the suffixes i, k, j, m may have any values from 1 to r.
Two sets of constants c^j, ... and c^^,... satisfying these
equations would not be considered distinct structure sets if
they could be connected by the equation system
h = r p = q = r
2* ahscihh = 2* aipakjcpqs>
where an, ... is a set of constants whose determinant
V
a
ii'
a
rH
a
ir
a
rr
does not vanish, as we explained in Chapter V.
Suppose however that, instead of being given the order
of the group, we are given the number of variables in the
operators of the groups, how are we to find all possible types
of groups in these variables 1 The method of finding the
structure constants is not now available ; for, when the number
of variables, n, is greater than unity, the order of the group, r,
332
CLASSIFICATION OF OPERATORS
[258
may have any value up to infinity. The problem suggested
has so far only been solved for the cases n = 1 , n = 2, n = 3.
In this chapter it will be shown how the groups of the straight
line, and the primitive groups of the plane may be obtained.
§ 259. A group Xlt ..., Xr, where
xk = &
+...+&»
gkl7ix1 *Knlxn
(k= l,...,r),
is transitive if it has n unconnected operators ; that is, if not
all w-rowed determinants vanish identically in the matrix
til!
fi,
Crl» • • • ferw
Now let »$, . . . , 05° be a point of general position, that is, a point
whose coordinates do not make all n -rowed determinants
vanish in the matrix, and in the neighbourhood of which all
the functions &*,"... are holomorphic. By transforming to
parallel axes through this point we may expand all the func-
tions &,■,..., in powers of xt, ...,xn; and we then see that from
the r operators of the group a set of n independent ones,
say X1, . .., Xn, can be selected such that
k *«k
+ 6ivr + ••• + &
(&= i,...,n),
where £u vanishes for xx = 0, ...,xn = 0.
The other (r — ri) operators of the group Xn+1, ...,Xr may
be so chosen that for each of them £,-• , when expanded, has no
term not beginning with powers of x1, ...,xn, that is, no con-
stant term. These (r — n) operators form a sub-group, the
group of the origin, characterized by the property of leaving
the origin at rest.
If in an operator
£ 1 -\ ™ + • • • + fen
^x, ' b" %x,
n
the lowest powers of xx, ...,xn which occur when £ls ..., £n are
expanded are of degree s, then we say that the operator is of
degree s.
If we have a number of operators F1? ..., Yq each of degree
s, and if no operator dependent on these, that is, of the form
Cj i j + ... + e^ J: g, »
260] ACCORDING TO THEIR DEGREES 333
where ex, ...,eq are constants, is of higher degree than s, we
say that they form a system of degree s. It is clear that we
cannot have more than n operators in a system of degree zero
nor more than n 2 in one of degree unity, and so on.
If then the operators Xn+X, ...,Xr do not form a system of
degree unity, we can deduce from them a number of operators
of the second degree ; and proceeding similarly with these
latter we may be able to deduce a system of the third degree,
and so on.
We therefore see that the operators of a transitive group
may be arranged as follows : n operators forming a system of
zero degree, mx forming a system of the first degree, m2 a
system of the second degree, . .., ms a system of sth degree.
Since all of these operators are independent, and the group
is finite, 8 cannot exceed a finite limit, and we have
r = 71 + 7)1-^ + ... +ms.
If we form the alternant of two operators of degrees p and
q respectively, it can be at once verified that it cannot be of
degree lower than^ + g — 1. This principle is of great use in
determining the possible types of groups when n is fixed ; we
shall now apply it to obtain the possible finite continuous
groups in a single variable, that is, the groups of the straight
line.
First, we notice that if a group contains no operator of
degree h, then it cannot contain one of degree (k+ 1) ; for it
must have, if transitive, n operators of zero degree, and, by
forming the alternants of these with the operators of degree
(k+ 1), we must have operators of degree k.
§ 260. We now consider the case where n is unity; we may
take the operators of such a group to be
^x x^x ^x -l>x ix *b+1*x
where ^ contains x in degree i at the lowest ; and in this
group there must be no operator of degree higher than s.
Suppose that s > 2 ; then, forming the alternant of the
operators of degree s and (s — 1 ) respectively, the group must
contain an operator of degree (2 s — 2), viz.
x2S~2 *x + &*-1 ^ '
which, since s > 2, would be an operator of degree higher than s;
334 THE POSSIBLE GROUPS [260
and, as this is impossible, we conclude that s cannot be greater
than two.
A group in a single variable cannot then contain more than
three independent operators.
A general principle, whatever may be the number of vari-
ables, is that all operators of the kih and higher degrees form
a sub-group. This is proved from the fact that any two such
operators have an alternant whose degree is at least (2k— 1),
and therefore not less than k, unless k is zero; if k is zero
the operators of the kth and higher degrees form the group
itself.
If from the operators X1,...,Xr we form a new set of
operators, by adding to any operator of degree k any operator
dependent on the operators of degree not less than k, we shall
still have the operators of the group arranged in systems of
degree zero to s. Advantage of this principle may often be
taken to simplify the structure constants of a group.
Thus in the case of a single variable, suppose 8=2, and let
X0 , X1 , X2 be the three independent operators respectively of
degrees 0, 1, 2. From the group property we have
(X1,X2)=aX0 + bX1 + cX2,
where a, b, c are constants.
Since (Xls X2) is of the second degree, a and b must be
zero ; and, by comparing the coefficients of r— on the two
sides of the identity, we see that c is unity.
Similarly we see that
(^o> ^2) = 2X1 + eX2,
where e is some unknown constant.
To eliminate this constant, we take as the operators of the
group T0,Y1, 72 where
F0 = X0, Y1 = X1 + ^eX2, Y2 = X2,
and we have
(1) (Y1,Y2) = Y2> (Y0,Y2) = 2Y1. }
Suppose now that
(2) (Y^YJ^Y^aY. + bY,,
where a and b are some unknown constants: from Jacobi's
identity
((F0, FJ, Yj + ((Ylt Y2), F0) + ((F2, YQ), Fj = os
261] IN A SINGLE VARIABLE 335
and therefore from (1) and (2)
a(Ylt F2) = 0J
so that a is zero.
We now take (/3 being an undetermined constant)
^0 ~ ^0 + ^-^2' %1 = ^1' Z2 = Y2,
and have
(Z0,Z2) = 2Zl, (Z1,Z2)=Z2, (Z0,Z1)=Z0 + (b-2p)Z2;
and therefore, by taking 2/3 = 6, we see that the group has
three operators Z0, Z1} Z2 respectively of degrees 0, 1, 2, and of
the structure
By a change of the variable from x to x' we can reduce
r — h A — to the form r— > ; to do this we have ~j~ = - — ■=■ 3
033 ^Tix <>x dx 1+&
where £2 is of degree unity in a? at least, and we may take x'
in the form a; +f(x), where f(x) is a holomorphic function of
x, whose lowest term is of the second degree in x at least. In
the new variables therefore Z0, Zx, Z2 will still be of degrees
0, 1, 2 respectively, but £x will be identically zero.
Omitting accents from the variable we take ZQ to be r— •
Smce (to' ^-^to^to'
we see that £2 must be a mere constant ; it must therefore
be zero, since it was given to be at least of the second degree
in x. We may similarly deduce that £3 is zero; and therefore
the only group of the third order is
a a 2 d
— J OJ^J flTr— •
d# C&* 003
Similarly we may see that the only group of order 2 is of
the type 3 s
and the only group of order unity is =— •
§ 261. Before applying this method to find the types of
groups in two variables, it will be convenient to consider how,
<>x„
336 SIMPLIFICATION OF ANY OPERATOR [261
by a liDear transformation of the variables, the operator
(1) (an a^ + ... + aln xn) — + . . . + (anl xx+...+ ann xn)
may be reduced to a simple form.
Let Aj be any root of the equation
= 0;
and let us find n quantities e1,...,en such that
aue1+ ...+ anl en = Ax e±
au-
•A, ft2i ,
•
• ani
a12
5 #22 ~~ "-J
•
• an2
•
•
•
• •
-
.
•
• •
<hn
' #2ra '
.
• ann — A
anlel+ ~- + ann en = Xien-
These quantities will, unless all first minors of the deter-
minant vanish, be proportional to the first minors of any row.
We take as a variable to replace some one of the set xx , . . ., xn ,
say xx, the expression y1 where
y1 = e1x1+... + enxn.
We then see that the operator (1) is of the same form in
the variables ylt x2,...,xn as it was in x1,,..,xn, but the
constants a y , ... are replaced by a new set of constants
ay, ... characterized by the property
a'n = Ax, a'12 = 0, ..., a[n = 0.
By a linear transformation, then, the operator (l) can be
reduced to such a form that
#11 = Aj, (Xj2 = 0, ..., (Xjjj = 0.
We similarly see that, by introducing a new variable y2
where
y2 — e2x2+ ... +enxn,
and e2, ...,en are determined by
#22 e2 + • • • + am en ~ A2 en>
a2n e2 + . . . -f ann en — A2 en,
the operator can be still further reduced to a form in which,
in addition to the former simplification, we have
#22 = A2' #23 == 0) •••s#9ra = 0.
5 «-2n
262] OF THE LINEAR GROUP 337
Proceeding thus we see that the operator can by linear
transformation be reduced to the form
(2) A^— + (a21x1 + \2x2)^-+(a31x1 + a32x2 + \3x3) — + ....
This operator may be still further simplified; suppose Ax
and A2 are unequal, and apply the transformation
2/i = xl> 111 ~ X2 + AXli 2/3 = *^3' '•'■>Vn~ xn
which gives
} d S d d d d
we then see that by a suitable choice of A, without otherwise
altering the form of (2), we can make the new a21 to be zero,
when we express the operator (2) in terms of the new variables.
Similarly, having caused a21 to disappear, by a transforma-
tion of the form
2/l = xl ' 2/2 = x2 > Vz == ^3 "f" A ^1 ' 2/4 == ^4' • • • ' Vn = *^n '
we could cause a31 also to disappear from the new form of the
operator ; and proceeding thus, so long as none of the co-
efficients A2, ..., Aw are equal to \t, we could cause an, ..., anl
to disappear.
In exactly the same manner, by properly choosing the trans-
formations, we could cause all the coefficients a&, ... to dis-
appear so long as none of the quantities A15 ..., Xn are equal ;
that is, if the determinant has no equal roots, the canonical
form of the linear operator is
11 ;\ ' ' 2 2 "\ * ""•••• "1" ^fi^fi
§ 262. The general method of obtaining a canonical form
for the case of equal roots will be sufficiently explained by
considering the case where Xx = A2 = A3 = A4 , and no other root
is equal to Ax.
First consider the coefficient of - — ; by the transformation
*x5
2/5 = x^ + ax^, y1 = ajj, ..., yn = xn
we can by a suitable choice of A cause a54 to disappear ; and
by a similar transformation we can cause a^, a52, a51 also to
disappear.
CAMPBELL Z
338 THE OPERATORS OF [262
It is thus seen that the operator may by a linear trans-
formation be brought to such a form that xx, x2, x3> x± only
appear in the first four terms.
These terms take the form
*» (* 4 + *24 + Xz 4 + Xi 4p + Xi ^ 4 + ctu 4 + a* 4}
+ ^32^2^ +1^42^2 + ^43^3/ ^ '
Now by any linear transformation in x1} x2, x3, x± the part
d 3 a c)
X-i r r flj,. r + flUo T + Xa r
1c)aj1 ^a;2 ^#3 ^«4
is unaltered ; if a21 is not zero by a transformation of the form
Vl = «!i 2/2 = X2> 2/3 = «*! 2/4 = ^4 + ^2
we can eliminate the new «41 ; we may then by a trans-
formation
2/l = aJH 2/2 = ^2. 2/.3 = «3 + Aa;2! 2/4=^4
eliminate a31 ; and then, if a32 is not zero, we may eliminate
ai2 ; while if «32 is zero by a transformation
2/l = X\i 2/2 == ^2 + ^-^3' 2/3 ~ X3> 2/4 == ^4
we may eliminate a43.
If a21 is zero, but not a32, we take
2/l = Xl> 2/2 = a32 ^2 + a31 ^1 ' 2/3 = ^3' 2/4 == ^4'
and thus eliminate a31 ; if a21 and a32 are both zero, but not
a^, we take
2/l = *^l ' 2/2 = ^2 ' 2/3 = a41 ^1 + a '42 *2 + ^43 ^3 ' 2/4 = ^4 »
and thus eliminate a41 and a42. Finally if a2l, a32, and ai3 are
all zero, we can similarly eliminate a41. Summing up we see
that the first four terms may be reduced to the form
, d 7) a 3 N
<)x2 L l7>x3 3 s^x±
■where elt e2, e3 are symbols for constants; and it is easily
seen that, by further simple transformations, we may reduce
263] THE LINEAR GROUP 339
these constants to such forms that any one, which is not zero,
is unity.
Similar expressions could be obtained for the other parts of
the operator ; and we thus see how, in any given number
of variables, to write down all possible types of such
operators.
We know of course that any linear operator can be reduced
to the type — ; but such reduction is not effected by a linear
transformation, and just now we are only considering how to
obtain types by linear transformation ; that is, types con-
jugate within the general linear homogeneous group.
§ 263. We now enumerate the types of linear homogeneous
groups of order one in two variables x, y ; we write p for
— and q for — » and e for an arbitrary constant :
^>x 2 c>y J
(1) e(xp + yq)+xp-yq, (2) xp + yq + xq,
(3) xp + yq, (4) xp-yq, (5) acq.
We shall now find all possible types of linear groups of the
third order.
First we find all the groups containing the operator
(3) xp + yq ; by a linear transformation every operator of
the group we seek can be reduced to one of the above five
forms (though the same transformation will not necessarily
bring two operators of the group simultaneously to these
normal forms) ; and a linear transformation cannot alter the
form of (3).
Since we only require two operators to complete the group
of the third order which contains (3) ; and, since these must
be independent of (3), one of the operators may be taken to
be of the form (4) or (5).
Suppose it is of the form (4), the remaining operator of the
group must be of the form
a {xp + yq) + b (xp—yq) + cxq + dyp,
where a, b, c, d are constants ; as we only require the part
independent of (3) and (4), we may take a and b to be zero.
Form the alternant of (4) with
cxq + dyp,
and we shall see that cxq — dyp
is an operator within the group. As the group is to be of the
z 2
340 THE HOMOGENEOUS LINEAR [263
third order, and to contain (3) and (4) ; and, as we now see
that cxq and dyp are operators of the group, we must have,
either d zero when the group is
(6) asp— yq, xp + yq, xq;
or c zero, when we get a group of the same type ; that is,
a group transformable into (6) by a linear transformation.
If we had assumed that the second operator was of the
form (5) we should have been led to the same group (6).
We must now find the linear groups of the third order
which do not contain the operator (3).
Suppose that one operator of our group is of the type (5) ;
and let a second operator be
a (xp + yq) + b (xp — yq) + cyp.
Forming the alternant with xq we see that the group will
contain c(xp_yq);
first we suppose that c is zero ; and we take the third operator
of the group to be
(7) a1(xp + yq) + b1(xp-yq) + c1yp,
where at, blt cx are constants.
Now cx cannot be zero, for, if it were,
a {xp + yq) + b (xp - yq) and ax {xp + yq) + bx (xp - yq)
would be two independent operators of the group ; and there-
fore xp + yq would be an operator of the group, which is
contrary to our hypothesis.
Forming the alternant of (7) and (5) we see that the group
will contain cx(xp-yq),
and therefore the group which contains (5), and does not
contain (3), must contain (4).
We therefore take the third operator of this group to be
a(xp + yq) + byp;
and forming the alternant with (4) we see that the group
must contain yp, and we thus have the group
(8) xq. yp, xp-yq.
We obtain the same group by supposing the first operator
to be of the type (4).
We have now only to find any possible group of the third
order which does not contain any operator of the types (3),
(4), or (5).
264] GROUPS OF THE PLANE 341
Suppose that one operator is of the type (2) ; we then take
a second to be
a(xp + yq) + b(xp-yq) + cyp,
and the third
«! (xp + yq) + 6j (xp - yq) + cx yp,
and we may clearly suppose that either c or cx is zero ; say
we take c to be zero, if we now form the alternant of
a (xp + yq) + b (xp — yq)
with (2), we shall get an operator of the type (5), which is
contrary to our hypothesis.
The group cannot therefore contain an operator of the type
(2); and we see similarly that it cannot contain one of the
type (1).
The only groups of the third order are therefore
xq, xp-yq, xp + yq,
and xq, xp — yq, yp.
It may be shown in a similar manner that the only groups
of the second order are
e(xp + yq) + xp-yq, xq ;
xp-yq, xp + yq;
xq, xp + yq.
We have now found all possible sub-groups of the general
linear group in x, y ; we might have obtained these directly
by the method explained in Chapter XIII.
§ 264. It is now necessary to examine the groups which
we have found ; and to see, with respect to each of them,
whether there is any linear equation
Xx + iiy — 0
admitting all the transformations of the group.
It may be at once verified that the group
xq, xp-yq, xp + yq
is admitted by the equation x = 0 ; that is, by any trans-
formation of this group, points on the line x = 0 are trans-
formed so as still to remain on the line x = 0.
It may similarly be proved by successively examining these
groups that, for each group, at least one linear equation can
be found to admit the transformations of that group, unless
the group is either
342 THE PRIMITIVE GROUPS [264
(1) the general linear group,
xq, yp, xp—yq, xp + yq,
or (2) the special linear group,
xq, yp, xp-yq.
§ 265. We now proceed to determine the types of primitive
groups of the plane.
If a group is imprimitive it must have at least one in-
variant equation of the form
g =</>(*, !/)•
We express this condition geometrically by saying that
an infinity of curves can be drawn on the plane ; and that by
the operations of the imprimitive group these curves are only
interchanged inter se ; any set of points, lying on one of the
curves of the system, being transformed so as to be a set,
lying on some other curve of the system.
If then we take a point of general position the group of
the point, that is, the transformations of the imprimitive group
which keep that point at rest, cannot alter the curve of the
system which passes through the point ; and in particular
the direction of the curve at the point is not altered.
•We take the origin to be a point of general position ; then
the lowest terms in the group of the origin are of the first
degree ; suppose P is the origin, and PT the tangent to any
curve which passes through P ; by the operations of the
group of the origin this curve will be transformed into a
system of curves all passing through P ; and the directions
of the tangents at P to these curves are what the direction
PT has been transformed into by the operations of the group
of the origin.
Now the only terms in the group which are effective in
this transformation of the linear elements through P are the
lowest terms ; that is, the linear elements at P are trans-
formed by a linear group.
We obtain this same result analytically as follows : —
let £+,,__
* 7>x t*y
be any operator of the group of the origin, so that £ and tj,
the terms of lowest degree in x, y, are at least of the first
266] OF THE PLANE 343
degree ; and let us extend the operator (denoting by p the
quantity -^) so as to get
tVx+r>k+{l1l+P{ri2~Q-p2Qk' ■
where the suffix 1 denotes partial differentiation with respect
to x, and the suffix 2 partial differentiation with respect to y.
We are only concerned to know how the p of any line
through the origin is transformed ; this we know through
the operator
where after the partial differentiations have been carried out
we are to take x = 0, y = 0 ; we therefore need only consider
those parts of £ and 77 which are linear in x, y.
Now if the group is imprimitive at least one value of p can
be found which is invariant for the group of the origin ; but
if the group is primitive no such value can be found. If
therefore the group is primitive the operators in it of the
first degree, according to the classification explained in § 259,
must either be of the form
3 d d d
(1) 2/5—+..., «—+..., #- y—+...,
v ' * dx cy dx dy
where the terms not written down but indicated by + . . . are
of higher degree in the variables than those which are written
down ; or else they must be of the form
a d a a a a
(2) yr+"-> x—+..., x^ y^-+..., x — +y^ + ...;
<$x oy dx oy dx ay
for, by § 264, all other forms for the group of the origin would
leave invariant at least one linear element through the origin.
§ 266. Suppose that the operators of the first degree are
of the form (1); it will now be proved that there cannot be
any operator of degree three, and therefore not any of higher
degree.
Suppose that there could exist in the group the operator
(i) »*5 + "-
344 THE PRIMITIVE GROUPS [266
where the terms not written down are of higher degree than
those written down ; form its alternant with
SC r r • • • ,
hy
when we shall see that the group must contain
Forming the alternant of (1) and (2) we get
(3) ,■£+...,
and forming the alternant of (2) and (3) we get
and so on ad infinitum ; so that the group would not be
finite as all of these operators are independent.
We can now prove that there can be no operator
(4) *S+'S5 + --'
where £ and r\ are of the third degree ; forming the alternant
of (4) with y — + ... we get
Forming the alternant of this again with y - — h . . . , and so
on, we get successively
(2/2£ii-22/Th)— +yvn^ +...,
(^m-32/2r7ll)-+2/r,m- + ...,
-42/3^n^ + ....
Now r;ni is a constant, and it must be zero, else would the
group have an operator
266] OF THE PLANE 345
and therefore 77 must contain y as a factor ; similarly we see
that £ must contain x as a factor.
We must now try whether there can be an operator of the
form
where £ and 77 are of the second degree ; forming the alternant
with y - — h ... we have
0 OX
(6) (yX^ + y^-rl))Vx+,fr]l~ + ....
Now the coefficient of — , being of the third degree, must
be divisible by x ; and therefore £— T7 must be divisible by x ;
by symmetry it must be divisible by y, so that
£-77 = axy,
where a is a constant.
The result at which we have arrived is that in any operator
of the third degree
, a a
£^ M a — r...j
^x oy
g-i-x — r)-±-y is divisible by xy. Applying this theorem to (6),
and writing 77 + ctxy for £, we see that a is zero, so that £ and
77 are equal.
We then have to try whether the group can contain an
operator of the form
v dx oy/
where £ is of the second degree.
Forming its alternants with the operators of zero degree
viz. - — l- .... and r — h ..., we obtain the two operators
Tix ^y
and forming the alternant of these two we have
a _^
346 THE PRIMITIVE GROUPS [266
This operator being of the third degree, must be such that
=2 _
-&
x y
and, £ being of the second degree, we must therefore have
iz = hx, & = — %,
where k is a constant.
NOW r- & = V" &
^2/ ox
and therefore & must be zero ; so that £ being of the second
degree and & and £2 both zero, £ must vanish identically.
We have therefore proved the theorem we enunciated, viz. that
no operator of degree three can exist in the group.
§ 267. We have now to find the possible forms of operators
of the second degree ; let such an operator be
First we could prove as before that the hypothesis of an
operator of the form ^
i>X
tfT- +
existing in the group would involve the non-finiteness of
the group.
Form successive alternants of (l) with y — +...; and we
get s
and therefore, since we must have rjn zero, we see that rj
contains y as a factor. Similarly we see that £ contains x as
a factor ; and we need only consider operators of the form
where £ and r/ are of the first degree.
Form the alternant of (2) with y — + ..., and we shall see
that £— t] is divisible by x, and therefore by symmetry it is
also divisible by y ; but £— >/ is of the first degree, and there-
fore must vanish identically.
268] OF THE PLANE 347
The only possible operators of the second degree are
therefore a %
where £ is of degree unity.
So far the reasoning has only involved the existence of two
of the operators of the first degree, viz.
(7) x - — H... and y f-...,
oy v ox
and it therefore applies equally to either class (1) or class (2)
of the primitive groups.
We now assume that the group is of the first class and
so has no operator of the form
and we shall see that £ must be zero.
Forming the alternants of
t( * *\
^xvx+y^>+-
with o , o
— +..., and — + ...,
ox oy
we have in the group the operators
<9> > I .*
b2 v ex " oy' h oy
Since £ is linear and equal, say, to ax + by, the existence
of (9) and (7) involves the existence of (8), unless a and b
are zero.
A primitive group of the first class can then only have the
five operators
d o o o o o
:r — + ..., :r — + ..., X - + ..., y r — + ..., X — — y r h ... .
ox oy oy ° ox ox oy
§ 268. We shall now for brevity denote by P the operator
y- — |- ..., by Q the operator x- — h ..., and by R the operator
o ^
x r y - + . . . .
oy oy
348 THE PRIMITIVE GROUPS [268
P, Q, R is the group of the origin, and we have
(P,R)=2P, (Q,E)=-2Q) (P,Q)=_p.
Also, since P, Q, R form with — + ... and — + ... the
group itself, dx °y
(ty +~" p) = chP+hQ+CiR+ ^ +...,
(^ +...,Q) = a2P+b2Q + c2R+ — +...,
where a1, 615 c1, a.,, b2, c2 are unknown constants.
If we now take as two operators of the group X and F
where
X=a1P + /31Q + y1R+^- +...,
Y=a2P + !32Q + y2R+ — + ...,
we get
(Y,P) = X+(a1-a1)P + (b1-p)Q + (c1-y1)R + (34Q,P)
+ y2 {&, -P)
= X+(a1-a1 + 2y.2)P + (h1-(31)Q + (c1-y1 + l3.2)R-
and, similarly,
(X,Q)=Y+(a2-a2)P + (b2-l32-2y1)Q + (c2-y2-a1)R.
We now choose the undetermined constants a15 /315 y15
a2' /^2> y2 80 as to make
(1) (F,P) = X and (X,Q) = F.
We next suppose (a2J62, ... denoting unknown constants)
that (Y,Q) = a2P + b2Q + c2R;
for obviously (Y,Q) does not involve X, Y, when we express
it in terms of X, Y, P, Q, R, a set of five independent operators
of the group which is of order five. Similarly we take
(X,P) = a1P + b1Q + c1R. |
We now apply Jacobi's identities to eliminate as far as
possible these unknown structure constants of the group.
From
(Q, (Y, P)) + (P, (Q, Y)) + (F, (P, Q)) = o,
(Q, (X, P)) + (P, (Q, X)) + (X, (P, Q)) = o,
269] OF THE PLANE 349
and from (1) we now have
(Y,R) = Y-b2R + 2c2P,
(X,R) = X+a1R-2c1Q-,
and from
(R, (Y, P)) + (P, (P, Y)) + ( Y, (P, R)) = 0,
we deduce
(RiX) + (P,b2R-2c2P-Y) + 2(Y,P) = 0;
that is, 2c1Q-a1R + 2b2P = 0,
which, since the operators are independent,
gives cx = a2 = b2 = 0.
Similarly we see that c2 = a1 = b1 = 0 ;
and we have now proved that
(Y,Q) = 0, (X,P) = 0, (Y,R) = Y, (X,R) = X.
In order to complete the structure of the group, we have
now only to express the alternant (X, Y) in terms of X, Y, P,
Q, R ; suppose that
(X, Y) = aX + bY+cP + dQ + eR;
from (P, (X, 7)) + ( F, (P, X)) + (X, ( Y, P)) = 0
we deduce that bX + dR — 2eP = 0,
and therefore b = d = e = 0.
Similarly we see that a and c are both zero, and the group
has therefore the same structure as the group
a S J) * o o
(2) — , ;— , yx—> x—, xs 2/ —
v ' ex oy ox oy ox oy
The group (2) and the required group are then simply
isomorphic, and the sub-groups of the origin correspond, so
that (§ 133) the groups are similar. The only primitive group
of the plane of the first class is therefore of the type (2) ; that
is, the type is that of the special linear group whose finite
equations are
x'= ax + by + e, y'= cx + dy+f,
where ad — be is equal to unity.
§ 269. We now have to consider the possible primitive
groups of the second class, when the group of the origin
contains
d d o o ^ o
y ox oy ox a oy ox oy
350 THE PRIMITIVE GROUPS [269
We have seen that the only operators of the second degree
are of the form
t(x^+y^)+~" i
X\
where £ is a linear function ; forming the alternant of this
with y - — h . . . , we get
ox
where £2 is a constant.
Similarly we see that the group must contain
y^xYx+y^)+"-
Unless then both £x and £2 are zero, that is, unless the
group contains no operator of the second degree it will contain
nxVx+y*y)+-~-
Similarly it will contain
f o o,
(x— +yT-) + ....
v ox oy/
If the group contains no operator of the second degree
it may be proved as before that it is of the type of the general
linear group
O O O 0 0 0 0 0
ox dy dy uox ox J oy ox J oy
If it does contain an operator of the second degree the group
contains the eight operators
_S_ j^ o o o I
*X + "" o^+~" y^+'"> Xo~y~+-~> X^~yty+->
xvx+yry+"" <a:^+^)+-' y(xvx+y^)+-'
§ 270. Let us denote these operators respectively by
(l) X, 7, P, Q, R, U, V, W.
We have at once (U, V) = V,
270] OF THE PLANE 351
since the alternant (U, V) being of the second degree cannot
involve X, Y, P, Q, or R.
So also (U, W) = W, and (U,P) = aV + bW,
where a and b are unknown constants ; and if we take instead
of P the operator P — aV — bW, we shall have
(U,P-aV-bW) = 0.
Since the lowest terms in P — aV—bW are the same as in
P, we may suppose that the operators (1) are such that (U, P)
is zero ; similarly we may suppose that ( U, Q) and ( U, R)
are zero.
We have
(U,X) =-X + aP + bQ + cR + dU+eV+fW,
which, by taking a new X with the same initial terms as the
original X, is reduced to
(U,X)=~X;
and similarly (IT, Y) = — Y.
Now by a change of coordinates we can transform any
linear operator into any other ; and in particular we can
transform
x— +y — + ... into x' ^j+y^-j
dx dy dx ,7 }>y'
by the transformation formulae
af=x + g, y'= y + rj,
where £ and 77 are functions of x and y, which, when expanded
in power series, begin with terms of the second degree at least.
If then we apply this transformation formula the lowest
terms in X, Y, P, Q, R, V, W will not be altered in form, U
will become x= — hV^-' and the structure constants will of
Tix oy
course be unaltered.
It will now be proved that
X = ~, F=* P = y~, Q = x±,
lx Zy u ^x * ly*
R==xvx-yvy> u=xTx+yvy>
V=x(xvx+yi^ w=y(xTx+yi^'
352 TYPES OF PRIMITIVE GROUPS [270
Take for instance
ox oy ox oy ox oy
where £(i) denotes a homogeneous function of degree k.
We have
and, as (?7, V) is equal to F, we must have
P>W r,(3> 1 + 2 (£(*>i- + ,,(*> A) +...
identically zero ; that is, £@\ tj(3), £W, rjW, . . . are all zero, and
V is merely x2 - — \-xy ^— -
Similarly for any other operator ; so that this primitive
group is of the type
oooooooo
— J X—> Xir V — 5
ox oy ox oy
o o o „ o
«r-' a; r— » x^ Vr-' # ^ — V y ■
ox oy ° ox oy ox oy ox oy
x2^ — i-ct^) xy - — I- y2 ;— >
ox ^ oy J ox J oy
that is, of the type of the projective group of the plane.
There are therefore only three types of primitive groups in
the plane, viz. (1) the special linear group ; (2) the general
linear group ; (3) the general projective group.
:, +VkT7.' (*=l,...,r),
CHAPTER XXII
THE IMPEIMITIVE GROUPS OF THE PLANE
§ 271. We shall now sketch the methods by which the
imprimitive groups of the plane may be obtained.
The group being imprimitive, the plane can have an infinity
of curves drawn upon it, such that by any operation of the
group these curves are only transformed inter se.
We therefore choose our coordinates so that these curves
will be given by x = constant, and then the linear operators
of the imprimitive groups must be of the form
where £ is a function of x alone.
If the operators of the group are now Xt, ..., Xr where
then it is clear that £, ^— , .... £,r—
^Zx r^x
must generate a group ; and, this being a group in a single
variable only, we can, by a change of coordinates (which
merely consists in taking as the new variable x' a certain
function of the old variable x) reduce £k to be of the form
ak + bkx + ckx2 where ak, bk, ck are mere constants. By
a change of coordinates the operators of an imprimitive group
can therefore be reduced to the form
Xk = (ak + bkx + ckx2)—+rjk — > (k= l,...,r).
It then follows that imprimitive groups of the plane can be
divided into four classes : the first class will only contain
operators in which ak, bk, and ck are zero, that is, they will
all be of the form rjj. — \ the second class will contain one
a * }y . ^
operator - — h 77, — , while all others will be of the form 77^. — :
dx xZy ^y
CAMPBELL A a
354 IMPRIMITIVE GROUPS [271
the third will contain the two operators
2> 3 c S
ox oy ox oy
with others of the form ri7. — - ; the fourth class will have
d 3 J) d _ 3 o
r h 77, ^— , # 1- ?72 — — , X- x. V r)3 —
ox oy ox oy ox oy
with others of the form 77 7. — — •
When we have found all possible forms of groups of one
class, in order to find the forms of groups in the class next in
order, we take one of these groups, and add to it the operator
which differentiates the higher from the lower class. Applying
the conditions for a group, we thus find the form of the operator
we have added, and the additional conditions necessary (if any),
in order that the group of lower class may thus generate one
of higher class ; this principle will be sufficiently illustrated
in what follows.
§ 272. We have first to find the groups of the form
c o
Since x now occurs merely as a parameter we can, by a trans-
formation of the form
x'=x, y' = f(x,y),
reduce each of these operators to the form
(«/, + /37,2/ + y7,2/2)^5
where a^ , (3^ , yk are functions of the parameter x only ; this
theorem follows from what we proved as to groups in a single
variable.
It may be at once verified that by a transformation of the
form , ~
v =■ — — — 3
J y + ty
where a, /3, y, 8 are functions of x only, any operator
(H+Pky+Yky)^
273] OF THE FIRST CLASS 355
is unaltered in form, the functions aj., /%, yk being trans-
formed into other functions of x. The operators of the group
are therefore unaltered in form by any transformation of the
given type.
Suppose that for every set of constants A15 ...,Ar the quad-
ratic function of y
Aj r/j + . . . -f Xr r\r
is a perfect square ; we may then assume that
Vh = H(ay + P)2> (k = l,...,r),
and therefore, if we take
we may reduce the operators of the group to such a form that
y does not occur explicitly in the group at all.
The first type of group that we find in this class is there-
fore of the form
« ['iw£ *,<•>£]■
Since all the operators are permutable, this group is an
Abelian one.
§ 273. We next consider the case where the operators are
all of the form
(ak + Pky)^> (k=l,...,r),
that is, the case where all the functions ylt ..., yr are zero ;
we cannot at the same time have all the functions /31S ..., (3r
zero, for then this type of group would reduce to the form
just considered.
Suppose therefore that /3X is not zero, and apply the trans-
formation y' = a1 + fi1y, which will enable us to take one of
the operators of the required group to be
Forming the alternant of this with (a2 + ^22/)v~ we ^n<^
that a2 /3j — is an operator of the group. Now if all the
functions a2, ..,, ar are zero we can by the transformation
y'= log y reduce the group to the type (1); we therefore
assume that a2 is not zero, and forming the alternant of
A a a
356 IMPRIMITIVE GROUPS [27
a* j3i — and B, v r— we find that a9 B,2 — is an operator of the
tH1*y * *y *y * a
group. Similarly we should see that a2 j8is j— » «2 Pi ^7 ' ' ' '
o if
are all operators of the group ; and therefore, if the group
is to be finite, we must assume Bi to be a mere constant, and
we may take this constant to be unity.
We may similarly show that all the functions 32, ..., 3r are
mere constants ; and we thus get the second type of groups
in the first class to be
<2> '1W5 F'-^w ȣ"
§ 274. We now pass to the case where there is at least one
function ax + 3X y + yx y2 which is not a perfect square and in
which y1 is not zero.
Let at + 31y + Yi y2 = y1(y- a) (y - 8),
and apply the transformation y = 3 which gives
We therefore again assume that the group contains an
operator B1y — ; and, if we are not to obtain the type (2)
over again, there must be at least one other operator
{a2 + B2y + y2y2)~
in which y2 is not zero.
By a transformation yf= y2y we may simplify the discus-
sion by having only to consider the case where y2 is unity.
Forming the alternant of (a2 + 8<,y + y2) — and Bx y — ,
a ■ *y *y ;
we find that (A 2/2 - a2 /3:) — is an operator of the required
°y ^
group. Forming the alternant of this again with 3xy — ,
y
and so proceeding, we get
W f + ft2 «2) ~ > W y2 - ft3 «2) ~> • • • ,
275] OF THE FIRST CLASS 357
so that the group would be infinite were not /3j a mere con-
stant, which we may take to be unity.
The group now contains
(2/2-a2)- and (y2 + aj~,
and therefore y2 — and a2 — ; forming the alternant of these
two we see that it contains a„ y — 3 so that aQ is a constant.
The group contains (a2 + /32 y + y2) — , and therefore also
d . *y
/i22/ — j so that j32 is also a mere constant.
If (a3 + /332/ + y32/2)— is any other operator we find, by
taking its alternant with y — 3 that the group will contain
(«3 + 732/2)^ and (y^f-a5)~ ,
and therefore a.3 r— » yq ?/2 — » and therefore also /3, v r— : and
^^2/ ^2/ ^2/
we see as before that a3, /33, y3 must be mere constants.
If a2, ...,ar are all zero the group will therefore be of the
type a 5
*y J*y
which is but a particular case of (2) ; but if they are not all
zero the group will contain the three independent operators
(3) sr-5 Z/^-' 2Tn-»
<>y <>y °y
and no others.
We have now found that all groups in the first class must
be of the types (1), (2), or (3).
§ 275. Passing to groups of the second class, and first
taking (1) of § 272, we have to find the conditions necessary
in order that
1X ' 2>y r v ' <>y <>% oy
may generate a group of order (r + 1).
If all the functions F1, ..., Fr vanish identically we can
358 FORM OF LINEAR OPERATOR [275
reduce - — l- r\ — to the form — by a change of coordinates,
ox oy ox &
and thus obtain the type ^
(4) W
If they are not all zero we form the alternant of — + n —
21 ox oy
and F1 (x) — j and thus see that the group contains
y^-^r- must now be dependent on ^ ^., and there-
fore — is a function of x alone.
oy
We then take r\ to be of the form ay + /3, where a and /3 are
functions of x ; and it may easily be verified that by a trans-
formation of the form
x'=x, y'=y<t>(%)+f(x),
we may reduce - — f- 77 — to the form — , without essentially
o JO v 'iJ QUO
altering the form of the group
We have therefore first of all to see what forms these
functions F,, ..., Fr must have in order that (1) and — -
OX
may generate a group of order r + 1 .
§ 276. We now make a short digression in order to consider
a principle of which much use may be made in the investiga-
tion of possible types of finite groups.
If X is any linear operator of the group which we seek,
we can by a change of coordinates reduce it to the form
— ; if then any other operator of the group is
.a a , o
q^- + v ^— + C;— + ...>
ox oy OZ
we see, by taking its alternant with r-— » that
oP 0 or, 0 oC 0
— 1 L 1 » |_
ox ox ox oy ox oz
276] OF A FINITE GROUP 359
is an operator of the group ; so also must every linear
operator of the form
d*£ 3 y-ri d a*o
— - — H 1- — h ...
<ixk 3# 2>xk 2>y %xk 3s
belong to the group.
Now the group being finite only a certain number of these
operators can be independent ; and therefore there must be
some operator of the form
(where a15 ..., aj are constants, depending on the structure
constants of the group, and m15 ..., nij. are positive integers)
which will have the property of annihilating each of the
functions £, 77, C, ....
It follows that
+ ea*x(a21xm2-1+a22xnu--2 +...) + ...,
where a^,... denotes a function of the variables not con-
taining x ; and that we shall have similar expressions for
f]j V>j • • • •
Since (s ax) £— + (t ax) rj— + (r- — %) f^-+...
is an operator within the group, which will not contain x in
a higher power than (m^ — 2) in the coefficient of eQlX, and
, 3 v* 3 ,3 x2 3 ,3 y* 3
v3cc ' 3# v3# v 3i/ v3ce J 3s
is an operator in which x only enters in the power (m1 — 3) in
the coefficient of eaiX, and so on, it is not difficult to see that
the group must contain the following sets of operators.
Operators in which
£=eaixaly r] = eaixb1, Q=^xcXi
£ = e& (axx + an), 7] = eaix {\x + bn), C = ^x (ci% + cu),
£ = eaix (ax x2 + 2anx + a12), v = ea^x (b± x2 + 2bnx + b12),
C = eaix (cxx2 + 2 cuaj + c12),
and so on, where the letters a15 619 c1} ... all denote functions
not containing x.
360 IMPRIMITIVE GROUPS [276
In addition to these there will be the similar sets of
operators corresponding to the roots a2, ..., aA. ; and every
possible linear operator of the group will be dependent on
the operators here enumerated.
§ 277. Applying this principle to the problem before us,
viz. the determination of -the forms of Ft, ..., Fr in order that
may be the operators of a finite group, we see that the functions
denoted by alt bx, cti ... are now mere constants ; and that the
group must therefore be of the form
(5) \j*x^-t xea*x^-,...,Xm*-1ea*x~, ~~\, (A= 1,2,8,...).
\_ °y oy dy dxj v ' ' ' "•/■
§ 278. We have now to find what groups in the second
class may be generated from
*«W^ F*(x)Vy> *f?
by adding the operator - — h 17— — -
Forming the alternant of F1 (x) — and r— + n — , we see that
dy dx dy
^y dx J dy
is an operator of the group ; and therefore
Jt = r
(A) liwg-^M.^+s^H
where c15 ...,cr, and c are absolute constants.
Similarly, by forming the alternant of y — and M — ,
dy dx dy
we see that b,b1,...,br being a set of constants
From (A) we see that ?; is of the form a + /3y + yy2, where
279] OF THE SECOND CLASS 361
a, {3, y are functions of x only; and irom (B) we see, on
substituting this value for r), that y is zero, and
k = r
Now without loss of generality we can add to ^ V r\ —
any operator dependent on "
and we may therefore suppose that the form of q is so chosen
that both a and y are zero.
By a transformation of the form
x'=x, i/ = y4>(x)
we may, without essentially altering the form of the other
operators of the group, so choose the unknown function <j) (x)
that ^+/32/4 m^beCOme ^;
and we may thus reduce the group to one of the type
(6) [
o o o o o 1
oy' oy' ' oy' J oy' ox
(k= 1,2,3,...).
§ 279. The only type of group in this class remaining to be
examined is
;> 3 2 o _3_ o
ty> y^f y }>y' ox + Vo~y"
o c 0 \
Forming: the alternant (—, x — I- 77 ^- — ) we see that, there
0 \oy ox oyJ
being only four operators in the group,
-5 - a + 2by + 3cy2,
oy
where a, b, c are mere constants ; and therefore
r] — $ (x) + ay + by2 + cyz.
Forming the alternants of \- r? — with y — - and y2 —
& ox oy d oy J oy
362 IMPRIMITIVE GROUPS [279
respectively, we see that (j>(x) must be a mere constant, and
C must be zero ; so that the group reduces to the type
/^ S ° 2 * *
() ^' ^' ^V ^'
§ 280. In the third class the groups must contain two
operators of the forms - \-ri, r— and x- I-t?o — : and clearly
r ox oy ox 'oy
in any group of this class there must be a sub-group con-
taining all the operators of the group except x - — h r/2 — •
We therefore begin by trying whether from the group
[«■**—, xea"x~, ..., ^%-ie«^A, llJ fa = 12,3,...),
L *y oy oy oxj v '
we can generate a new group of order one higher, by adding
an operator of the form x - Yt\x—'
ox oy o
Forming the alternant of the new operator with — we see
-x OX
that — is a function of x only ; and forming its alternant with
V 77
any other operator of the group we see that — is a function
of x only ; and therefore we take
V = cy + <j>(x)
where c is a mere constant.
If we substitute this value of ?? in a? - \-r> — , and form the
alternant with x™k 1eQkX~, we shall find that the group
must contain ak xmkeakX — ; and, as xmk~1 is given to be the
highest power of x in the coefficient of eakX, we conclude that
a.], must be zero.
The group must therefore be of the form
c c o c c ,3
r-s X—+r)--, —t X — , ..., a?-1 — »
ox ox oy oy oy oy
where r\ = cy + 2 c hxJc + constant ;
281] OF THE THIRD CLASS 363
and without loss of generality we may say that
■q = cy + crxr.
If c is not equal to r, apply the transformation
/ / CrX
x = x, y = y + — — t
c — r
when the group takes the simple form
, N a d o o o r . a
(8) — , a — +C2/ — , — , x — ,...,cc'-1 — •
' ox dec d?/ d?/ ay oy
If c is equal to r it is easily seen that by a transformation
of coordinates we may take cr to be unity, and thus obtain
the type
,M J d a d d ■ a
v ' ox ox v J 'oy oy oy oy
§ 281. We should next have to try what groups of the form
Td d n d d d d dl
\eakx xeakx xmk-leakx y x +r] ,
\_ oy oy oy ° oy ox ox oy_
{k= 1,2, 3,...)
can exist ; and in much the same way we should see that we
may take rj to be cxy when c is a constant. If we then apply
the transformation
x'=x, y' = e~cxy,
— becomes r— ; — cvf — , > V =— is unaltered in form, and
ox ox <>y °y
x uv — becomes #'— ,> whilst the other operators are not
ox oy ox
essentially altered in form. If we now apply the same
reasoning to this type as we applied to the last, we shall see
that ak must be zero, and that the group takes the form
Td d „ , d oo d"i
(10) — , x— , ..^x*-1 — , y^> ^3 x^-\>
ivy <>y ^y ^y ox ox^
where r > 0.
The other types of group in this class can similarly be
found ; they are
r d d „ d d d"i
IMPRIMITIVE GROUPS
(12)
r a * an
— » x — + — ;
(13)
L^a;' bx]'
[281
§ 282. Passing to types of groups in the fourth class we
must take each group from the third class, and see whether
we can generate a group of the fourth class by adding to
it some operator of the form x2 \-v — •
bx by
Thus it may easily be shown that from
r-» x — , ...,xr J— i — , a; — -+(r?/ + af)— , where r>0,
a group of the required class cannot be generated. On the
other hand, the group
d b . , 2> d 3 d
03/ t>2/ Sy bx bx ^ d£/
will lead to two types of group of the fourth class ; viz.
,xT^ d « , d d o Nd „d
(14) — , x—,...,xr-1 — ,—,2x—+(r-l)y — , x2—+(r-l)x%
Iby by by dec da; v d2/ da;
where r is greater than zero ; and
L d^/ da; da; da; ^ d?/J
The other types of groups in this class are
(16) —, x— , ...jo;'-1 — > 2/ — > — » oj— > a;2— + (r— l)aw — > (i
[by by by J by bx bx bx v ' by]
' \by by by bx bx bx]
(18) |— , X—+y — , x1 — + (2 xy + y2) — 1 ;
v ' \bx bx Jby bx v J byy
/.«x r d a s . d d-i
(19) — , 2x--+y-~s x2—+xy— ;
7 Ua; da; ^ d*/ da; J by] '
a
0
2 *1
0T —
- — i
OJ — j
OX
diC
<)#
283] OF THE FOURTH CLASS 365
(20) [;
The methods by which these groups of the fourth class
are found does not differ essentially from the methods by
which the groups of lower class were found.
§ 283. Every imprimitive group of the plane must belong
to one of the types enumerated, but these types are not all
mutually exclusive ; thus the group
o d . o
oy Joy J oy
in the first class is similar to the group of the fourth class
r— 3 X — ■> £T — •
ox ox ox
In order to divide the imprimitive groups into mutually
exclusive types we examine each of the groups we have found
as regards their invariant curve systems. For all the groups
the system x = constant is an invariant system, but some of the
groups have other invariant curve systems.
We first consider the type (1) and suppose that r is greater
than unity ; we may then by a transformation of coordinates
of the form
x'=x, y'=y4>{x)
simplify the type so as to be able to assume that two operators
of the group are — and x — •
* r oy oy
Suppose that for this group / (x, y) = constant is an in-
variant curve system ; we must then have
— f(x, y) = some function off(x,y).
%j
If this function vanishes identically / (x, y) is a mere function
of x, and therefore only gives the known invariant system,
x = constant. If, however, the function does not vanish iden-
tically the curve system / (x, y) = constant can be thrown into
o f
such a form that ~- is unity, and therefore
oy J'
y +f (x) = constant
is an invariant curve system for the group. Applying the
366 THE INVARIANT CURVE SYSTEMS [283
operator x — of the group we must then have
x — (2/ +/ (#)) — some function of {ij +f (x)) ;
and as this is impossible we conclude that, if r is greater than
unity, (1) cannot have any other invariant curve system than
x = constant.
If, however, r is equal to unit)7, the group is of the type
— ; and admits the co°° curves y = f (x) as invariant systems,
where / is an arbitrary functional symbol.
We next can prove that if the type (2) is of order two.
it may be thrown into the form
& »41!
0
a
0
0 0
w
ox
or
ly'
<ix ^ oy
and for either of these groups there is an infinity of invariant
curve systems, viz.
ax + by = constant,
where a and b are arbitrary constants.
The type (6), if the order is three, can be thrown into
the form
a* a
dx oy ° dy
with the invariant systems x = constant, y = constant ; if the
order is above the third the only invariant system is x = con-
stant.
and for this group there are two invariant systems, viz.
x = constant, and y = constant. If the group is of order
greater than two the only invariant system is x = constant.
It will be found that for type (3) there are the invariant
systems x = constant, and y = constant.
The type (4) is similar to type (1), when the latter is of
order unity.
If the type (5) is of order greater than two, the only
invariant system is x = constant. If the group is of order
two it can be reduced to one or other of the forms
283] OF THE IMPRIMITIVE GROUPS 367
The type (7) has the invariant systems x = constant,
y = constant.
The type (8), if r>l, has only the invariant system
x = constant. If, however, r = 1 , the type is
3 3 3 3
— j —j x — +cy — ;
dx dy dx oy
and, since the group contains — and — , the invariant curve
' b r dx dy
system must be of the form
ax + by = constant ;
if c is equal to unity this system is admitted ; but if it is not,
the only systems admitted are x = constant and y = constant.
The group (9) has only the invariant system x = constant.
The group (10) has only the invariant system x = constant,
if r > 1 ; but, if r = 1, it has x = constant, y = constant.
The group (11) has the invariant systems x = constant,
y = constant.
The group (12) is similar to one of the cases of (5), viz.
the case when (5) can be thrown into the form
3 3 3
dy dx oy
The group (13) is similar to (2), when (2) is of the second
order.
The group (14), when r > 1, has only the invariant system
x = constant ; when r = 1 , it is
3 3 3 9 3
;— > ^— > X — > X — j
dy CX oX OX
and is similar to (7).
The group (15) has only the invariant system x = constant.
The group (16), when r > 1, has only the invariant system
x = constant ; when r = 1 it is similar to (11).
The group (17) has the invariant systems x = constant,
y = constant.
The group (18) is similar to
3 3 3 3 3 9 3
- — h — 3 x- — h y — , xl - — i-f-j
dx oy dx dy dx oy
and has the invariant systems x = constant, y = constant.
368 MUTUALLY EXCLUSIVE TYPES [283
The group (19) has only the invariant system x = constant.
The group (20) is similar to (3).
§ 284. We now rearrange the imprimitive groups of the
plane into mutually exclusive types and into four new classes,
corresponding to the different systems of curves, which are
invariant under the operations of the groups. We shall denote
the operator £ — + tj — by ip + yq.
In Class I we have the group q for which an invariant
system is y+f(x) = constant, where f(x) is any function of x
whatever.
In Class II
[q,p]; [q,xp + yq]; [q, p, xp + yq] ;
with the invariant curve systems
ax + by = constant,
where a and b are any constants.
In Class III
fa. 2/?]; fa. y?. »■?] ; \j>>q>yq];
fa) 2/<7. 2/2#' P] > fa> P> xP + cyq], c being a constant not unity ;
fa. yq> p, wp\ ; fa, yq, y2q, p, &p] ;
fa, yq, y2q, p> ®p, ®2p] ; [v + q, ®p + yq, %2p +y2q];
with the invariant curve systems x = constant, y = constant.
In Class IV
[Fx(x)q, ...,Fr(x)q], where r > 1 ;
[Fi(®)q, ...,Fr(x), yq], where r > 1 ;
\ea^q, ..., xnik~1 eak%q, p], where the order > 2, and k= 1, 2, 3, J
[eakXq, . . ., xmk~l eauXq, yq, p], where the order > 3, and &= 1, 2,3, .]
[q, xq, ..., xr~^q, p, xp + cyq], where r > 1 and c is a constant ;
• [q, xq, ..., xr~1q, p, xp + (ry + xr) q], where r > 0 ;
[q, xq, ..., xr~lq, yq, p, xp], where r > 1 ;
[q, xq, ...,xr~1q, p, 2xp + (r-l)yq, x2p + (r—l)xyq],
where r > 1 ;
284] OF IMPRIMITIVE GROUPS 369
[q, xq, ...,xr~1q, yq, p, xp, x2p + (r — l)xyq], where r > 1 ;
[yq,p, xp, x2p + xyq];
[p, 2xp + yq, x2p + xyq\ ;
with the invariant curve system x = constant.
It is clear that a group in one class cannot be similar to
a group in any other class ; and it may easily be seen that in
the same class no two similar groups have been enumerated.
Every imprimitive group of the plane must therefore belong
to one of these twenty-four mutually exclusive types.
CAMPBELL
B b
CHAPTER XXIII
THE IRREDUCIBLE CONTACT TRANSFORMATION
GROUPS OF THE PLANE
§ 285. We have now found all point groups of the plane,
and if we extend these we shall have all the extended point
groups ; if the groups are only extended to the first order and
we apply to them contact transformations we shall have the
reducible contact groups of the plane. In this chapter we
shall show how the irreducible contact groups of the plane
are to be obtained.
It must first be proved that the necessary and sufficient
condition that a system of contact operators of the plane
may be reducible to mere extended point operators by a con-
tact transformation of the plane is that the operators should
leave unaltered an equation system of the form
dx dp __ dy
a fi ap
where a and /3 are functions of x, y, p.
Let f(x, y, p) = constant, <f> (x, y, p) = constant
be integrals of this equation system ; then, since
<>x ly1 ^pa ^x %yx ^pa
we see, by eliminating - > that the functions / and 4> are in
involution ; we can therefore find a contact transformation
(1) x'=f{x, y, p), y'= $ (x, y, p), p'=$ (x, y, p)
which will transform the given equation system into
dx'- 0, dy'= 0.
Now if £=^ +»Jt- +wc-
dsc <>y ^p
be a contact operator which leaves unaltered the equation
system dx = 0, dy = 0, we Bee that £ and ij must be functions
286] CONDITION FOR REDUCTIBILITY 371
not containing p ; and therefore the operators, as transformed
by (l), will be mere extended point operators. The converse
is easily proved ; for extended point operators do not alter the
equation system dx = 0, dy = 0 ; that is, they transform a
point M1 into a point M^. It follows that if we apply to
them a contact transformation the reducible operators will
leave unaltered the equation system into which dx = 0, dy = 0
is transformed ; that is, an equation system of the form
dx _ dp _ dy
a (3 ap
§ 286. We now take x and z as the coordinates of any point
in the plane, and we write y instead of /;>, when the contact
operators of the plane become simply those operators in space
#, y, z which do not alter the equation
dz — ydx = 0.
An irreducible group of contact operators of the plane,
when regarded as operators in space, must be transitive. For,
suppose the group is intransitive, and/(«, y, z) is an invariant:
then the operators of the group do not alter the equations
^-dx + J-dy + J-dz = 0, dz — ydx = 0.
7>x oy oz
They therefore leave unaltered a system of equations of the form
dx dy _ dz
a ~ p ~ ay'
and therefore may be so reduced as to be mere extended
point group operators.
Let £(«, y, z) — + v (x, y,z)^~+ Cfa V>z)^>
or, as we shall write it
ip + vq + Cr,
be a contact operator of the plane regarded as an operator in
space x, y, z ; and let W be its characteristic function, so that
f 7>W *W iW _ IW
<>y ' lx J hz b J ly
Taking a point of general position as the origin of co-
ordinates, we can arrange the operators of the group into sets
b b 2
372 THE OPERATORS OF [286
as in § 259. To do this we expand the characteristic function
in powers of x, y, z ; let W be the operator which corresponds
to the characteristic function W, that is, let
ly1 ^^x J Zz '^ ^ J ZyJ
We must, therefore, in order to obtain an operator of degree
k, consider the terms in W which are of degrees (k + 1 ) and k.
Thus corresponding to W— — 1 we have W = r, and corre-
sponding to W = — x we have W = q + xr; more generally we
may express these, and similar results, in the tabular form
W= (-h ( — a; , Cy, ( -« , ( -z? , S xy ,
W= ( r , \q + xr, \p, (yq + zr, (2xq + x2r, \xp — yq,
W= ( y* , ( -032! , C yz , ( -02 .
W= \2yp + y2r, \(z + xy) q + xzr, (zp—y2q, I2yzq + z2r.
This table gives us the operators corresponding to terms in
W of the second or lower degrees, and, if required, could
easily be extended so as to give the corresponding operators
for terms of higher degree. Thus, if W= a + bx + cxy, where
a, b, c are constants, then
W= —ar — b(q + xr) + c (xp — yq).
It will be noticed that the only terms in W which contribute
operators to W whose lowest terms are of zero degree are
1 , x, y ; and the only terms which contribute operators of the
first degree are
z, x2, xy, y2, xz, yz.
The most general contact operator of the first degree is
therefore
(1 ) ax(yq + zr) +a2xq + az (xp — yq) + aiyp + a5zq + aez2)+ ...,
where ax, ...,aG are constants, and the terms indicated by
+ . . . are of degree higher in x, y, z than those written down.
§ 287. If we have a contact group, and consider the operators
of the first degree in the group, we have, by neglecting the
terms in such operators indicated by + . . . , a group which is
linear and homogeneous in x, y, z. From the form given by
(l) of § 286 for these operators, we see that the plane z = 0 is
invariant under the operations of this linear group ; the straight
lines through the origin in this plane are therefore transformed
288] THE FIRST DEGREE 373
by the operations of a linear homogeneous group in x, y.
Unless, then, this linear group is the general or special linear
homogeneous group, it must leave at least one straight line
through the origin at rest ; and therefore the contact group
itself must, when we regard it as a point group in space, leave
unaltered at least some oo2 curves which pass through co3 points
of space ; the considerations which enabled us to determine
the primitive groups of the plane will render this evident.
Now a contact group with the property of leaving co2 curves
at rest has been proved to be reducible ; and therefore the
linear group must be either the general or special linear
group.
The group we are investigating must therefore contain at
least the following three operators of the first degree
(1) yp + a1zp + b1zq + ...i
(2) xq + a2zp + b2zq+ ...,
(3) xp— yq + a3zp + b3zq + ....
Since the alternant of the first two of these operators is of
the form xp — yq + a3zp + b3zq + ..., it will only be necessary
to assume that the group contains the first two operators.
From the form of the general contact operator of the first
degree ((1) § 286), we see that there cannot be more than six
independent operators of the first degree, such that no operator
of the second degree is dependent upon them ; and since the
group is transitive in x, y, z there must be three of zero
degree. We have therefore to consider four possible classes
of groups ; in each there will be the three operators
JJ i • • • j \l i • • • j * "T" • • • 3
in Class I there will be three operators of the first degree ; in
Class II four such operators ; in Class III five, and in Class IV
there will be six.
§ 288. We first examine the possible forms of irreducible
groups in Class I; since the three operators (1), (2), (3) of
§ 287 must occur there cannot be any operators of the forms
zp + . . . , zq+ . . . , or yq + zr +
If we form the alternant of (1) and (2) we get
(y + a1z)q—(x + b3z)p + ... ;
and therefore, adding (3), we see that by the limitation im-
374 IRREDUCIBLE CONTACT GROUPS [288
posed on this class we must have (a3 — b2) zero, and also
(h + ai) zero- Similarly, by forming the alternants of (1) and
(3), and of (2) and (3) respectively, we see that a2 and 6X are
both zero.
The operators of the first degree in this class are therefore
(x + az)q + ..., (x + az)p-(y + bz)q+..., (y + bz)p+...,
where a, b, c are constants ; and it will now be shown that
there are no operators of the second degree in any group of
this class, and therefore no operators of any higher degree.
By the point transformation in space
(A) x'=x + az, y/=y + bz, z'=z
the operators of zero degree, and of the first degree, can be
thrown into the forms
p + ... , q + ... , r+... ,
xq+..., xp—yq + ..., yp+,„.
It will be noticed that this transformation is not a contact
transformation of the plane.
Suppose now that the group could contain an operator of
the second degree
€p + *)q + (r+...,
where £ rjt £ are homogeneous functions of the second degree
in x, y, z.
If we form alternants of this operator with p+ ..., q + ...,
r+..., respectively, the resulting operators, being of the first
degree, must be dependent on xq+..., xp — yq+..., yp + ,..,
and operators of higher degree ; and therefore the first deriva-
tives of £ ?/, C cannot contain z ; it follows that the functions
£, t], ( themselves cannot contain z.
Also, since there is no operator of the first degree in which
the coefficient of r is not zero, the derivatives — and — are
~&x ^y
both zero ; and therefore ( vanishes identically.
If, then, any operator of the second degree is to be found in
the group at all it must be
(B) £p + riq+-..,
where £ and 77 are homogeneous functions of the second degree
in x and y.
There can, however, be no such operator ; for we proved in
§ 267 that the operators
p+..., q + ..., xq+..., Xp-yq + ..„ yp+...
289] OF THE FIRST CLASS 375
could not coexist in any finite group with an operator of the
form (B), unless the group also contained the operator of the
first degree nuin , a,„ .
& xp + yq+ ... ;
and, as the group we are now considering does not contain
this operator, we draw the conclusion that in Class I there can
be no operator of the second degree, and therefore none of
higher degree.
§ 289. The group has therefore only six operators ; for
brevity we denote
p+... by P, q+... by Q, r+... by R,
xq+... by X15 xp-yq + ... by X2, yp+... by X3.
Clearly in this group Xx, X2, X3 is a sub-group — the group
of the origin ; its structure is
(X15 X2) = — 2XX, (X15 X3) = X2, (X2, X3) = — 2X3.
We also have
(X1; P) = -Q + axXx + bxX2 + cxX3,
(X2,P) =-P + a2X1 + 62X2 + f2X3,
(X3, P) = azXx + 63 X2 + cz X3,
where ax, bx, cx, ... denote constants.
By adding to P and Q properly chosen multiples of Xx, X2,
X3, we may throw these structure constants into the simple
form
(X1,P) = -Q, (X2,P)=-P, (Xz, P) = a,Xx.
If X, F, Z are any three linear operators we know that
(X, (F, Z)) + (F, (Z, X)) + (Z, (X, F)) = 0 ;
this Jacobian identity may be written in the abbreviated form
(X, F, Z) = 0.
From (Xls X2, P) = 0, we now deduce that (X2, Q) — Q ;
from (X3, Xx, P) = 0, we similarly obtain (X3, Q) = — P;
while from (X2, X3, P) = 0, we shall find that a3 is zero.
The alternant (Q, Xx) is dependent on Xx, X2, X3 ; if then
(Q,Xx) = aXx + bX2 + cX„
we deduce from (Xl5 X3, Q) = 0 that a and b are zero ; while
from (X2, Q, Xx) = 0 we shall see that c is zero, and therefore
(Q, Xx) is zero.
376 IRREDUCIBLE CONTACT GROUPS [289
If we now apply the transformation inverse to (A) of § 288,
V1Z- x = x' + az', y = y' + bz*, z — z',
we shall bring the operators of the group back again to such
a form that they are contact operators of the plane x' ', z' ; and
we may therefore say that the group in Class I has the six
operators
r p + ..., q+..., r + ...,
(x + az)q + ..., (x + az)p — (y + bz)q, (y + bz)p+....
If we denote these respectively by P, Q, R, X19 Z2, Z3, we
now know so much of the structure of the group as that
(Xv X2) = — 2Xlt (Z15 X3) = X2, (X2, X3) = -2Z3,
(l) (Zv P) =-Q, (X„ P) = -P, (X„ P) = o,
(ZVQ) =0, (X2,Q) =0, (X3,Q) =0.
§ 290. If we now form the alternant of P and Q it will be
of the form
r + ap + (3q + ...,
where a and /3 are constants. For, if u and v are the character-
istic functions of the operators u and v, the characteristic
function of the alternant (u, v) is
^/(^ + y^-^fx+yYz)-UVz+VJ-z;
and, as the lowest terms in the characteristics of p + . . . and
q+ ... are respectively y and — x, the lowest term in the
characteristic function of their alternant must be — 1 , and
therefore the lowest terms in the alternant must be of the
form r + ap + /3q.
We may then say that
(P,Q) = R + aP + l3Q + yX1 + bX2 + eX3,
where a, /3, y, 8, e are constants ; and we may therefore so
choose an operator R as to have (P, Q) = R without altering
the structure of the group in so far as it is given by (1)
of § 289.
From the identity (X1} P, Q) = 0 we then see that (X13 R)
is zero ; and we similarly obtain (X2, R) = 0 and (X3, R) = 0.
We now take
(P, R) = a1P + 51Q + ClE + a1Z1 + ^Z2 + y1Z3,
where ax, blt c15 a15 /3X, y1 are constants.
291] OF THE FIRST CLASS 377
From (X3, P, R) = 0, we see that ax, blt f31 are all zero;
from (X2, P, B) = 0, we see that cx and y1 are zero; while
from (X15 P, R) = 0, we see that
(X1(P,R)) + (Q,R) = 0.
We therefore have
(P, R) = aP, (Q, R) = aQ, (P, Q) = R ■
and from (P, Q, R) = 0, we now deduce that a is zero.
The structure of the group is now given by
(P,Q) =R, (R,P) =0, (Q,R) =0,
(X15 P) = -Q, (Z2, P) = -P, (X3, P) = 0,
(1) (Xv Q) = 0, (Z2, Q) = Q, (X3, Q) = -P,
(XltR) =0, (X2, P) = 0, (X3,P) =0,
(X2, A3) = — 2X3, (X?j, Xj)= —X2, (X15 X2) = — 2X1.
§ 291. In this group the operators P, Q, R form a simply
transitive sub-group of the same structure as the simply
transitive group whose operators are
p, q + xr, r ;
it is therefore possible to find a point transformation which
will transform P, Q, R to these respective forms.
If we take X15 X2, X3 to be (in the new coordinates thus
introduced) respectively
£iP+vi2+(ir* £2P+v2<i+(2r> €3P+v3q+Czr>
then, from the structure constants of the group, we derive
a number of equations which these functions £ls r}x, (t, ...
must satisfy.
It will be at once seen, on forming these equations, that
they will be satisfied by taking
£i = °> Vi — x, Ci = I %\ 4 = a. V-i = V> C2 = °>
£3 = 2/> ^3 = °> C3= hy2;
and therefore a possible form of group is
(1) p, q + xr, r, xq + lx2r, xp — yq, yp + \y2r.
Now any group in Class I can be reduced to such a form
as to have the structure given by (l) § 290 ; and for such
a group X15 X2, X3 will be the sub-group of the origin. The
most general group of the class we seek is therefore simply
378 OTHER CLASSES OF [291
isomorphic with (1); and in this isomorphism the groups of
the origin correspond, so that (§ 133) we conclude that the
most general group is similar to (1) ; that is, it is reducible
to the form (1) by a point transformation in space x, y, z.
§ 292. It must finally be proved that this point transforma-
tion is a contact transformation in the plane x, z.
First it may be seen that (1) of § 291 is a contact group,
and that it satisfies the condition of irreducibility ; we see
that all the operators are contact operators, since the cor-
responding infinitesimal transformations do not alter the
equation dz — ydx = 0 ; and we conclude that the group is
irreducible because the lowest terms in the operators of the
first degree form the special linear homogeneous group (§ 287).
Now suppose that the point transformation, which trans-
forms the general contact group of Class I into (1) of § 291
has transformed the Pfafhan equation dz — ydx = 0 into some
equation of the form
gdx + rjdy + Cdz — 0.
The group (1) of § 291 must therefore leave unaltered this
equation, and also, since the group is a contact one, it must
leave unaltered the equation dz — ydx= 0; but this would
necessitate that (1) of § 291 should leave unaltered a system
of the form dx_dy_dz
a /3 ay
where a and j3 are functions of x, y, z ; and therefore it would
be reducible, which we know it is not.
We conclude, therefore, that the only group in Class I is
that one which is reducible to
p, q + xr, xq + \x2r, xp-yq, yp + \y2r,
by a contact transformation of the plane.
§ 293. We shall now briefly consider the groups of irre-
ducible contact transformations of the other classes.
Every such group contains the three operators
(1) yp + a1zp + b1zq+...,
(2) xq + a2zjJ + b2zq + ...,
(3) xp—yq + a.dzp + bszq + ...;
and must contain at least one operator of the form
(4) a(xp + yq + 2z7*) + bzp+czq+ ....
293] IRREDUCIBLE CONTACT GROUPS 379
If we form the alternants of (1), (2), (3), (4) we see that the
group must contain the six operators
(1.2) (y + a1sr)q-(x + b2z)p + .,.;
(l, 3) (y + a1z)p-b1zq + (y-b3z)p+...;
(1,4) — az(a1p + b1q)—czp + ...;
(2.3) -2xq — (b2 + a3)zq + a2zp+ ... ;
(2.4) —az(a2p + b2q) — bzq + ...;
(3, 4) — az (a3p + b3q) — bzp + czq + ....
Now if the group is of Class III or Class IV it contains at
least one operator for which a is zero ; and therefore we see
from (1, 4), (2, 4), (3, 4) that it must contain zp+...} and
also zq +
If then the group is of Class III, as it can have only five
operators of the first degree, its operators must be
yp+..., ccq+..., xp — yq+..., zp+..., zq+....
If the group is of Class IV it has six operators of the first
degree, which must then be
yp+..., xq+..., xp — yq + ..., xp + yq + 2zr...,
zp+ ..., zq+ ....
It only remains then to find the operators of the first
degree for a group in Class II which can only have four
operators of the first degree.
For a group of this class a cannot be zero ; for then there
would be at least five operators of the first degree, viz. in
addition to (l), (2), (3), the operators zp + ..., and zq+ ....
From (2, 3), (3, 1), and (1, 2) we see that, since the group
contains (1), (2), (3), it must contain
(ai + h) zq + (a3 — b2)zp+..., 3 bxzq + (a1 + b3) zp+ ...,
3a2z}) + (b2 — a3)zq + ... ;
and therefore, since the group, being in Class II, can contain
none of these operators, we must have
ax + b3 = 0, a3 — b2 = 0, 6X =0, a2 = 0.
From the equations (1, 2), (l, 3), (2, 3), (1, 4), (2, 4), (3, 4)
we then deduce that
aax + c = 0, ab2 + 6 = 0, aa3 + 6 = 0, ab3 — c = 0 ;
380 IRREDUCIBLE CONTACT GROUPS [293
and, since a is not zero, it follows that the operators of the
first degree in Class II must be of the form
(x + az)q + ..., {x + az)p — (y + bz)q+..., {y + bz)p+..„
xp + yq + 2 zr — azp — bzq,
where a and b are some undetermined constants.
§ 294. Having found the initial terms in the operators of
the first degree, the methods by which we find the groups in
the Classes II, III, and IV are not essentially different from the
methods employed in finding the group in Class I, and in
finding the primitive groups of the plane ; we shall therefore
merely state the results which one will arrive at by such
an investigation.
Every group of Class II is reducible by a contact trans-
formation to the type
p, q + xr, r, xq + \x2r, xp — yq, yp+\y2r, xp + yq + 2zr.
In the third class no irreducible group can exist.
In Class IV every group is reducible by a contact trans-
formation to the type
p, q + xr, r, xq + \x2r, xp — yq, yp + \y2r,
xp + yq + 2zr, {z — xy)p—\y2q — \xy1r, \x2p + zq + xzr,
(xz — ixPy) p + (yz- \xy2) q + (z2 — \x2y2)r.
There are, therefore, only three types of irreducible contact
groups in the plane.
CHAPTER XXIV
THE PRIMITIVE GROUPS OF SPACE
§ 295. It would occupy too much time to attempt to
describe all the types of group which may exist in three-
dimensional space, and we shall therefore confine our
attention to the primitive groups which are the most in-
teresting. It will be shown that there are only eight types
of such groups.
The first theorem which it is necessary to establish is that
every sub-group of the projective group of the plane must
have either an invariant point, an invariant straight line, or
an invariant conic.
Suppose that u = 0 is a curve which admits two independent
projective operators X and Y, where
X=(P1 + xRl)±c+(Ql + yR1)~,
Y = (P, + xR.2)~ + (Q2 + yR2) A,
P15 Qx, i?15 P2, Q2, R2 denoting linear functions of x and y.
Then, since all points on the curve u = 0, must satisfy the
equations Xu = 0, Yu = 0 these points must also satisfy the
equation
Px + xRx, Q1 + yR1
P2 + xR2, Q2 + yR2
which, it is easily seen, is not a mere identity.
Now this is the equation of a curve of the third degree at
most, and, as it contains the curve u = 0, that curve is an
algebraic curve of degree three at the most.
§ 296. We shall now prove that this curve if a cubic must
be a degenerate one.
It is easily seen that if A, B, C, D are four points, no three
of which are collinear, there is no infinitesimal projective
= 0,
382 CURVES ADMITTING TWO [296
transformation which can leave all of these points at rest.
To prove this, we take any other point P on the plane, then
the pencil of four straight lines A (B, C, D, P) must be trans-
formed into a pencil of four other straight lines ; and if A, B,
C, D were to remain at rest, and P become transformed to P*,
we should have
A (B, G, D,P) = A (B, C, B, P'),
so that P' would lie on A P. Similarly it would lie on BP,
and therefore P' would coincide with P ; that is, every
point in the plane would remain at rest, which is of course
impossible.
Let A be one of the points of inflexion which every cubic
must have : if the cubic admits any projective group the
group must leave A at rest ; for an inflexion can only be
transformed to an inflexion, and therefore if A did not remain
at rest there would be an infinity of inflexions.
If the cubic has no double point it must have nine points
of inflexion ; and at least four of these points are such that
no three of them are collinear. A non-singular cubic cannot
therefore admit a projective group ; for the group would then
leave four non-collinear points at rest, which is impossible.
We conclude, therefore, that the cubic has a double point.
Suppose that it contains one double point and no cusp ; it
has then three points of inflexion, and these points, together
with the double point, must remain at rest under the opera-
tions of the group. But if a point A and three points B, C, D
on a straight line not passing through A remain at rest, the
only projective transformation which the figure could admit
would be a perspective one with A as centre and BCD as
axis of perspective.
An infinitesimal projective transformation cannot therefore
transform the cubic into itself ; for, if P is any point on the
curve and A the double point, P would have to be trans-
formed to a near point P' on the line AP ; and P' could not
be on the curve, since AP only intersects the cubic on
A and P.
Suppose now that the cubic has one cusp only ; since by
hypothesis the cubic admits at least two infinitesimal trans-
formations, there must be at least one infinitesimal transforma-
tion which will not alter the position of some arbitrarily
assigned point P on the cubic. From P draw the tangent
PQ which touches the cubic at a point Q distinct from P:
there will now be four points, viz. P, Q, the point of inflexion,
and the cusp which will not be altered by the projective
297] PROJECTIVE TRANSFORMATIONS 383
infinitesimal transformations admitted both by the point P
and the cubic itself. As we can so choose P that no three
of these points are collinear, we must conclude that the cubic
cannot be a proper one.
Since the cubic must be degenerate we conclude that the
only curves, which could admit a projective group with at
least two operators, are straight lines or conies.
§ 297. Any sub-group of the general projective group of
the plane must be either primitive or imprimitive ; we first
take the case where it is primitive, and therefore of one of
the two following types:
p, q, xq, xp-yq, yp, xp + yq;
p, q, xq, xp-yq, yp.
The first of these is the general linear group
xf=a1x + b1y + c1, y' = a2x + b2y + c2',
and it is clear that by any operation of this group a point
at infinity will be transformed to a point at infinity ; and
therefore the group leaves the line at infinity at rest. The
second group, being a sub-group of the first, must therefore
also leave the line at infinity at rest.
It now remains to prove that every imprimitive projective
group of the plane will leave either a point, a line, or a conic
at rest.
First we take the case where the group is at least of the
third order. From the imprimitive property of the group
we know there is an infinity of curves forming an invariant
system. If we take any one of these curves there must be at
least two infinitesimal transformations of the group which it
will admit ; for there are at least two such transformations
which will not transform any chosen point on the curve from
off the curve. Each of these curves must therefore, since
the group is projective, be either a conic or a straight line.
If the invariant system of co1 curves are conies, the five
coordinates of the conic must be connected by four equations,
and therefore the system of conies must have an envelope.
This envelope may consist of mere isolated points ; thus the
envelope of conies of the system u + kv = 0, where & is a
parameter, consists of the four points of intersection of the
two conies u = 0 and v = 0.
Similarly, if the invariant system of x1 curves are straight
lines, they must have an envelope.
Now the envelope is invariant under all the transformations
384 INVARIANTS OF A SUB-GROUP [297
of the group ; and, if it does not consist of a mere set of
isolated points, it must therefore, by what we have proved,
be either a straight line or a conic.
A sub-group of the general projective group, if of at least
the third order, will therefore leave at rest either a point,
a line, or a conic.
We now suppose the sub-group to be of order two ; and
take Xl and X2 to be its operators ; we have
(X15X2) =aXx + bX2
where a and b are constants ; and therefore if we take as the
operators of the group X1 and aXx + bX2, we see that the group
must have the structure
(X1SX2) = &X2.
If b is not zero, by taking the fundamental operators of
the group (i.e. those in terms of which the others are to be
expressed) to be t Xx and X.2 , we have the structure
(X15X2) = X2;
if, however, b is zero the structure is given by
(X15X2) = o.
If the group is intransitive there will be an infinity of
invariant curves ; and, by what we have proved, these must
be straight lines or curves. If on the other hand the group is
transitive we throw X2 into the form — ; and then we may
■v ^ oX
take X, in the form x - — \- — » if the structure is given by
1 <*« ^y & j
(Xls X2) = X2 ; if the operators are permutable, we take Xx in
the form — •
ly
In either case the line at infinity is invariant under the
operations of the group ; and therefore returning to the
original variables some curve admits two infinitesimal pro-
jective transformations, and therefore must be either a straight
line or conic.
Finally if the projective group contains only one operator,
let it be
(ex + e2x + e3y + x (e±x + e5y)) p + (e6 + enx + e8y + y (e^x + esy))q.
The condition that the straight line
Xx + ixy + v = 0
298] OF THE PROJECTIVE GROUP 385
may be invariant requires it to coincide with
A (e1 + e2x + e3y + x (e±x + esy))
+ tJ.(e6 + e^x + esy + y(e^x + e5y)) = 0.
The equations therefore to determine A, fx, v are
Xe2 + [xe7—vei = k\, Xe3 + fj.es — ve5 = k[x, ke1 + ne6 = kv,
where k is to be determined by
&2 n?5 7 ' ^4
63 ' e8 "'J e5
6j , 6q , AC
0;
and there is therefore at least one straight line which the
group leaves at rest.
In every case, therefore, a sub-group of the general projective
group of the plane must leave at rest either a point, a straight
line, or a conic.
§ 298. We now proceed to show how the primitive groups
of space are to be obtained. We take as origin a point of
general position, and arrange the operators of the group
according to degree, as in § 259.
There will be three operators of zero degree
£>+..., q+..„ r + ...,
where we write v for — , q for — , r for — ; and a number of
1 <ix oy oz
operators of the first degree which cannot exceed nine. Let
the operators of the first degree be X1,X2, ... where
Xk=(aklx + ak2y + ak3z)p + (bklx + bk2y + bk3z)q
and akl, ..., bkl, ..., ckl, ... denote constants.
If we put x = uz' , y = vz' , z = z' , then in the new variables
the terms of lowest degree in Xk are transformed into
(aklu + ak2v + ak3-(cklu + ck2v + ck3)u)^
+ (bklu + bk2v + bk3-(cklu + ck2v + ck3)v) —
+ (cklu + ck2v + ck3)z — ,.
CAMPBELL Q Q
386 THE PRIMITIVE GROUPS [298
If we now regard u, v as the line coordinates of straight
lines through the origin, we see that the cc2 linear elements
through the origin are transformed hy the group of the origin,
in exactly the same way as the straight lines u, v are trans-
formed by Fls F2, where
Yk = (aklU + ak2V + ah3-(CklU + Ck2V + Ckz)U)^-
+ (bklu + h]r,v + bJc3-(chlu + Cj{2v + ckJv)—-
The linear operators Yx, F2, ... are now the operators of
a projective group in the variables u, v, and there cannot be
more than eight independent operators in such a group.
If there are eight independent operators F15 ..., F8 the group
is the general projective one
d d 3 d d
u— , u — , v—, v — , — >
OU OV dU OV OU
o 2 i , o o _ a
dW du OV du dv
and the terms of lowest degree in Xx, ...,XS are the terms of
the special linear homogeneous group
zp, zq, xq, xp — zr, yq — zr, yp, xr, yr.
It may be proved by the method of Chapter XXI that in
this case the primitive group we seek must be one of the
following three : —
The general projective group of space
n) [p> ?» r> xp> yp> zp> w> yq> z^ xr> vr> zr>
x2p + xyq + xzr, xyp + y2q + yzr, xzp + yzq + z2 r] ;
the general linear group
(2) [p, q, r, xp, yp, zp, xq, yq, zq, xr, yr, zr];
the special linear group
(3) [P, ?» r> «?» xp-y<l, yp, zp, zq, xp-zr, xr, yr].
§ 299. If Fl5 F2, ... are not the operators of the general
projective group they must form a sub-group of it; and must
therefore have the property of leaving at rest either a point,
a straight line, or a conic.
They cannot leave any point at rest ; for, if they did, the
group of the origin, viz. Xlt X2, ... and the operators of higher
299] OF SPACE 387
degree, would leave at rest a linear element through the origin,
and therefore the group would not be a primitive one.
Suppose that YX>Y2, ... have as invariant a straight line,
then the primitive group we are seeking must have an
invariant equation of the form
adx + fidy + ydz = 0
where a, /3, y are functions of x, y, z.
By a change of variables we can reduce this equation to the
form 7
dz — ydx = 0 *,
and the group we seek must therefore in the new variables
be a contact group in the plane xz.
If this contact group were reducible, it would have an
invariant equation system of the form
dx _ dy _ dz _
a ' ' /3 ~ ay'
and therefore, regarded as a point group in space, could not
be primitive.
Since then it must be irreducible, it can by a contact trans-
formation of the plane be reduced to one of the three forms :
(1) p, q + xr, r, xq + \x2r, xp — yq, yp + \y2r;
(2) p, q + xr, r, xy + \ x2r, xp — yq, yp + \y2 r, xp + yq + 2zr;
p, q + xr, r, xq + \x2r, xp — yq, yp + \y2r, xp + yq + 2zr,
(3) (z — xy)p—\y2q — \xy2r, \x2p + zq + xzr,
{xz — \x2y)p + (yz — \ xy2) q + {z2 — \ x2y2) r.
If a group is imprimitive, it must be admitted by some
equation of the form
(4) ip + yq + Cr=0.
Now if for a transformation of the form
(5) x'=f{x,y), y'=4>(x,y), z' '=^{x,y,z)
the equation (4) is invariant, then for the same transformation
the equation .
ip + vq = 0
must be an invariant one.
The group (1) can only be admitted by (4), if £, t;, £clo not
* It could not reduce to the form dz = 0, for then the group would be
imprimitive.
C C 2
388 THE PRIMITIVE GROUPS [299
contain x or z ; for only equations of this form could admit
the operators p and r. Again it is clear that every trans-
formation of (1) is of the form (5), and therefore
£p + r1q = 0
must admit the group
p, q, xq, xp-yq, yp,
formed by omitting the parts of the operators involving r.
This group, however, in x, y is primitive, and cannot be
admitted by an equation of the form £'P + f)<l = 0 ; and there-
fore we conclude that the only equation which could admit
(1) is the equation r = 0.
It can be at once verified that this equation admits both
the group (1) and the group (2), so that these groups are
imprimitive.
If the group (3) is admitted by an equation of the form
(4) £p + vq + Cr;
then, since (1) is a sub-group of (3), the group (l) must also
have the equation (4) as an invariant one ; from what we
have proved therefore, £ and rj must both vanish identically,
and we have only to try whether r = 0 admits the group (3).
Now it can be at once verified that it does not do so ; so
that (3) is the only primitive group of space obtained from
the supposition that Tx, T2, ... have as invariant a straight line.
§ 300. If we transform to the variables
y = y'2, % = -;> z = z\
then in the new variables the Pfaffian equation
dz — ydx= 0 becomes dz' —y'dx' + x'dy' '= 0;
and we have the primitive group of space x, y, z,
/■i\ P — yrf q + ocr, r, xq, xp — yq, yp, xp + yq + 2zr,
' zp — y (xp + yq + zr), zq + x (xp + yq + zr), z(xp + yq + zr),
characterized by the property of leaving unaltered the equa-
tion
dz—ydx + xdy— 0,
and transforming the straight lines of this linear complex
inter se.
301] OF SPACE 389
§ 301. We have now only to consider the case where
Yx , Y2, ... has an invariant conic which does not break up
into straight lines.
By a projective transformation any conic can be reduced
to the form 2 ,
ar + 2T + 1 = 0 ;
and we need therefore only consider the projective group
which such a conic can admit.
If the conic admits
(*! + e2x + e3 y + x (e±x + e5 y))p + (e6 + e^x + e8y + y(eix + e5y)) q,
we must have
e^ + etj = 0, e2=0, e8 = 0, e1—ei = 0, e5-e6 = 0 ;
and therefore the operator must be of the form
eiX + e2Y+e3Z,
where X = yp — xq, Y — (l + x2)p + xyq, Z = xyp + (l+y2)q.
The operators Yx, Y2, ... must therefore be the operators of
the group X, Y, Z with the structure
(7, Z) = X, (Z, X) = Y, (X, Y) = Z,
or of one of its sub-groups.
If the sub-group is of order one we have proved that it
leaves a straight line at rest, and therefore comes under the
case already considered.
Next we take the case where the sub-group is of order two,
and we take its operators to be
e1X + e2Y + e3Z and e1X + e2Y+e3Z.
Since the alternant of these two operators must be dependent
on them we must have
(eiX + e2Y+e3Z, €lX + t2Y+e3Z)
= p(e1X + e2Y+e3Z) + q(e1X + e2Y+e3Z);
and therefore, since the alternant is easily proved equivalent
(e2 e3 - e3 e2) X + (e3 e1-e1e3)Y+ (ex e2 - e2 ej Z,
we have
62e3 63e2' 63el 6le3' 6le2 62 el
^15 ^2 ' 3
el 5 €2 ' e3
= 0;
390 THE PRIMITIVE GROUPS [301
that is, (e2e3-e3e2)2 + (e3ei-eie3)2 + (eie2— e2ei)2 = °-
If we choose \, n, v to satisfy the equations
\e1 + fxe2 + ve3 = 0, \e1 + ixe2 + v€3,
it can be at once verified that the straight line
A = ixy — vx
admits this sub-group, so that this also falls under the case
already considered.
We have therefore only to consider the case where the
group Y1, Y2, ... is of the third order.
§ 302. We must now find the form of a group in x, y, z
which is of at least the sixth order, with three operators
of zero degree, and at least three of the first degree, and with
the property of having an invariant equation of the form
( 1 ) adx2 + bdy2 + cdz2 + 2fdydz + 2gdzdx + 2 hdxdy = 0,
where a, b, c, f, g, h are functions of x, y, z such that the
discriminant ahc + 2fgh _ af2 _ bg2 _ ck2
is not zero.
The equation (1) is not altered in form by any point trans-
formation, and it may easily be proved that by a suitably
chosen transformation we may reduce it to the form
(2) adx2 + bdy2 + cdz2 = 0.
The origin being a point of general position, and the dis-
criminant not being zero, we know that if we expand the
functions a, b, c in powers of the variables the lowest terms
will be of degree zero ; and by a linear transformation we
may take these lowest terms each to be unity. We must now
find all possible forms of primitive groups of order not less
than six which the Mongian equation (2) can admit.
Arranging the operators according to degree, as in § 259,
we shall first prove that the group cannot contain an operator
of degree three, and therefore none of higher degree.
If the equation admits the operator
ox oy dz
we must have, for all values of x, y, z, dx, dy, dz, satisfying (2),
2 a (€xdx + £2dy + £3 dz) dx + 2b (t/x dx + r)2dy + t\zdz) dy
+ 2c (C-^dx +C2dy+ C6dz) dz + Xa. dx2 + Xb . dy2 + Xc .dz2 = 0,
where suffixes are used to denote partial derivatives.
303] OF SPACE 391
It therefore follows that we must have
and, if p denotes some undetermined factor,
2a£1 + Xa = pa, 2brj.z + Xb = pb, 2c(3 + Xc = pc.
We now suppose Z to be an operator of the third degree
of which the terms of lowest degree are
sothat X = ^+v^ + C~+....
The equations satisfied by £, 77, f are now
^ + C2=0, Cl + &=°> ^2 + ^1 = °>
2a£ = pa, 2bih = pb, 2c(3 — pc,
since we may neglect Xa, Xb, Xc, as containing no terms
of degree less than three, while the derivatives of f, »/, £ only
contain terms of the second degree.
These equations can be written
V3+C2 = °> Ci + 4=°> 4 + ^1 = °> ^i = r?2 = C3;
and we have proved in Chapter II, § 35, that no values of
£, 77, £ of the third degree can be found to satisfy these equa-
tions ; we therefore conclude that the group cannot contain
any operator of the third degree.
§ 303. Still making use of the results of Chapter II, we
shall see that the only possible operators of the second degree
are dependent upon
(1) (^2_02)i_ + 2^ + 2^+...,
(2) 2xy—+(y2-z2-x2)— + 2yz~ + ...,
v ' a Ix ^y i>z
(3) 2zx — + 2yz—+(z2-x2-y2) — + ....
v ' Zx ?>y J^z
Similarly we see that the only possible operators of the first
degree are dependent upon
v ' ° i*z <>y
392 THE PRIMITIVE GROUPS [303
THE PRIMITIVE GROUPS
00
0
z ^r
ox
0
— x — + ..
oz
• 5
(6)
0
x —
oy
0
ox
• }
(?)
0
x —
ox
+ 2/— + z
ty
oz
and therefore the group is of the tenth order at highest.
We next see, as in § 264, by aid of the isomorphic group
F15 F2, ... in the variables u, v, that there must be three
operators of the first degree at least, viz.
o o,o o o ,
y- z — + e(x — + y — + z— )+...,
° oz oy v ox oy ozJ
o o,o o o v
z- x— +e[x—-+y— + z— )+...,
ox oz ^ ox oy oz'
o o,o o K
x- y -— + e (x — + y —- + z ~) + ...,
oy J ox ^ ox u oy oz'
where e is a constant.
If we form the alternants of these three we see that, unless
e is zero, the group must also contain
ooo
ox oy oz
and therefore the group must contain (4), (5), (6), and may
also contain (7).
If we denote by 1 the operator (1) and so on, we see that
1, 2, and 3 are commutative ; and that
(T, 4) = 0, (T, 5) = -3, (1, 6) = 2, (2, 5) = 0, (2, 4) = 3,
(2, 6) = -T, (3, 6) = 0, (3, 5) = I, (3, 4) = - 2.
From these identities we see that if the group admits any
operator of the second degree, viz. (1), (2) or (3), it must admit
all three.
We first consider the case where the group admits no
operator of the second degree, and not (7), but only (4), (5),
(6) in addition to the three of zero degree.
If we denote 4 by X, 5 by F, 6 by Z, and the three opera-
tors of zero degree,
P T • • • j y T • • • j 9* T • • • j
303] OF SPACE 393
by P, Q, R respectively, we have
(Y,Z) = -X, (Z,X) = -7, (X,Y) = -Z.
We also have, since X, 7, Z, P, Q, R generate a group,
(P, X) = ^X + b.Y + ^Z, (P, Y) = ~R + a2X + b2Y+c2Z,
(P,Z) = Q + a3X + b3Y+c3Z,
where ax , bx , ... denote structure constants ; if we add to
P, Q, R operators dependent on X, F, Z, we may throw these
identities into the simpler forms
(P,X) = aX, (P,Y) = -R, (P,Z) = Q}
where a is some constant.
From the Jacobian identity
(P, (X, F)) + (F, (P, X)) + (X, (F, P)) = 0,
which we now write in the form (P, X, F) = 0, as we shall
have occasion to employ it often, we deduce
(R,X)=-Q + aZ;
while, from (P, X, Z) = 0, we have
(Q,X) = R + aY;
and, from (P, F, Z) = 0, we have
(P, Z) + (Q, Y) = aX.
We now have (Q, Z) = - P + axX + bJ'+^Z,
and deduce, from (Q, X, Z) = 0, that
(Q, Y)-(R,Z) = c1Y-b1Z; and therefore
2(Q, Y) = aX + ^Y-^Z, 2(R, Z) = aX-^Y+^Z.
From (Q, Y, Z) = 0, we then conclude that a, ax, and cx are
zero ; and have so far determined the structure of the group
that we may say that
(P, X) = 0, (P, F) = -R, (P, Z) = Q, (Q, X) = R,
(Q,Y)=-bZ, (Q,Z) = -P+2bY, (R,X) = -Q, (R,Z) = bZ.
From (Q, X, F) = 0, we now see that
(P, F) = P-6F;
and, from (R, X, Y) = 0, we see that b is also zero.
Suppose that
(P, Q) = a1P + b1Q + c1R + kX + H.Y+vZ;
394 THE PRIMITIVE GROUPS [303
we then see from (P, Q, X) = 0, and from (P, Q, Y) = 0, that
(R,P) = c1Q-b1P-fiZ+vY,
(Q,R) = cxP-axR-\Z + vX;
and, from (P, Q, Z) = 0, we conclude that ax, bx, A, fx are all
zero, and therefore
(P,Q) = cxR + vZ, (Q,R) = c1P + vX, (R,P) = cxQ + vY.
If we now take as the operators of the group instead of P
the operator P + eX, instead of Q the operator Q + eY, and
instead of R the operator R + eZ, it is seen that the only-
structure constants which are changed are cx and v which
become respectively cx — 2e and v — cxe + e2. By properly
choosing e we can therefore throw the structure of the group
into the form
(Y,Z)=-X,(Z,X)=-Y,(X,Y)=- Z,(P,X) = o, (Q,Y)= 0,
(R,Z)= 0,(P,Y)=-R,(P,Z)= Q,(Q,X)=R, (Q,Z)=-P,
(R, X)=-Q,(R,Y)= P, (Q, R) = cP, (R, P) = cQ, (P,Q) = cR.
§ 304. Two cases now present themselves according as c is,
or is not, equal to zero.
First we take the case where c is zero.
P, Q, R now form a simply transitive Abelian sub-group.
By a point transformation we can therefore reduce P, Q, R
to the forms — > — > :— - respectively ; suppose that
where in £, 77, £ the lowest terms, when expanded in powers of
x, y, z, are of the first degree. From
(P,X) = 0, (Q,X) = R, (R,X) = -Q,
we see that (denoting partial differentiation with respect to
x, y, z by the suffixes 1, 2, 3, respectively)
£1 = vi = Ci = °> & = ^2 = °> (2 = 1> 6$ =(3= °> % = — 1 ;
and therefore X = y 0 — •
Similarly we see that Y= Zr x r— and Z = x~ y-;
J da; t»^ Si/ ^ ^x
and therefore the group is simply the group of movements in
ordinary space ; and the invariant Mongian equation is
dx2 + dy2 + dz2 = 0.
305] OF SPACE 395
Next we take the case where c is not zero ; and we choose,
as the fundamental operators of the group,
P, Q, R, X-cP, Y-cQ, Z-cR,
which we may denote by
P, Q, R, P', Q', R'.
The structure is now given by
(P, Q) =-R, (Q, R) =-P, (R, P) =-Q,
(P', Q') = -R\ (Q\ R) = -P', (R, P') =-Q',
while each of the operators P, Q, R are commutative with
each of the operators P', Q', R'.
We may also rearrange these operators, taking
U = ~P+iR, V=iQ, W = -P-iR,
U'=-P' + iR', V'=iQ\ W/=-P/-iR/,
where i is the symbol for V — 1 ; the group is now the direct
product of two simply transitive reciprocal groups.
Since U, V, W is simply transitive, and has the same
structure as
q + xr, yq + zr, (xy — z) p + y2 q + yzr,
it may be transformed into the latter when U', V, W will
be transformed into
p + yr, xp + zr, x2p + (xy—z)q + zxr.
It will be noticed that in this form the origin is no longer
a point of general position ; and it may at once be verified that
in this form the group has the invariant Mongian equation
<h'2 + y'2dx2 + x2dy2 + {±z—2xy) dxdy — 2xclydz — 2ydzdx = 0.
This group, which is admitted by the quadric z — xy = 0, is
the group of movements in non-Euclidean space.
§ 305. If we were to consider the case of a group containing
no operators of the second, but four of the first degree, and
three of zero degree, we should similarly obtain the group of
order seven consisting of movements in Euclidean space and
uniform expansion, viz.
p, q, r, yr — zq, zp — xr, xq — yp, xp + yq + zr.
Finally, if we were to consider the group containing three
396 THE EIGHT TYPES [305
operators of the second degree, we should find that there
must be four operators of the first degree in the group, as
well as three of zero degree ; and should arrive at the con-
formal group in three-dimensional space, consisting of move-
ments in Euclidean space, uniform expansion and inversions,
viz. the group
(1) [p, q, r, xq — yp, yr — zq, zp — xr, U, 2xU—Sp,
2yU-Sq, 2zU-Sr],
where U = xp + yq + zr and 8 — x2 + y2 + z2.
This group has the property of being the most general
group for which the equation
dx2 + dy2 + dz2 = 0
is an invariant.
By the operations of this group any sphere is transformed
into a sphere, and in particular any point sphere
(x-a)2 + (y-b)2 + (z-c)2 = 0
is transformed into some other point sphere. If, therefore, we
apply the contact transformation with the generating equations
x' + iz' + xy'—z = 0, x (x' — iz') + y — y'= 0*
by which spheres in space xr, yf, zf are transformed to straight
lines in space x, y, z, and point spheres to straight lines of
the linear complex
(2) dz + ydx—xdy = 0,
we should expect to obtain the projective group (1) of § 300,
for which the linear complex (2) is an invariant.
It may be verified that this is the case, and therefore the
groups (1) of § 300 and (1) of this article have the same
structure.
§ 306. We have now found all possible types of primitive
groups of space ; that all these eight groups are primitive is
easily proved ; the groups (1), (2), and (3) are primitive because
they have no invariant linear element for the group of the
origin, a point of general position; the group (1) has been
proved primitive ; and the groups (5), (6), (7), and (8) are
* These are obtained from the equations of Chapter XVII by the
substitution ,/= ^ s,= _,/i; x,= ^
x = -Xj, y = yl} s = -Sj.
306] OF PRIMITIVE GROUPS 397
primitive because the three operators of the first degree do
not leave any linear element through the origin at rest.
Collecting the results of this chapter we conclude that every
primitive group of space is of one of the following types :
(!) [p. ?. r> xp> yp> zp> xq> yq> zq, xr> vr> zr>
x2p + xyq + xzr, xyp + y2q + yzr, xzp + yzq + z2r];
(2) [p, q, r, xp, yp, zp, xq, yq, zq, xr, yr, zr] ;
(3) [p, q, r, xq, xp-yq, yp, zp, zq, xp-zr, xr, yr] ;
(4) [jj — yr, q + xr, r, xq, xp — yq, yp, xjD + yq + 2 zr,
zp — y(xp + yq + zr), zq + x (xp + yq + zr), z(xp + yq + zr)];
(5) [p, q, r, yr-zq, zp-xr, xq-ypJ\;
(6) [q + xr, yq + zr, (xy—z)p + y2q + yzr, p + yr, xp + zr,
x2p + (xy — z)q + zxr\ ;
(7) \jp, q, r, yr — zq, zp — xr, xq — yp, xp + yq + zr]-,
(8) [p, q, r, xq — yp, yr — zq, zp — xr, U, 2xU—S.p,
2yU-S.q, 2zU-S.r],
where U = xp + yq + zr and S = x2 -f y2 + z2.
CHAPTER XXV*
SOME LINEAR GROUPS CONNECTED WITH HIGHER
COMPLEX NUMBERS
§ 307. In this chapter we shall explain briefly an interesting
connexion between the theory of higher complex numbers
and that of a particular class of linear homogeneous groups.
k as i = n
(1) Let x's =2 asik xiVk> (s = 1, ..., n)
be the finite equations of a simply transitive linear group,
characterized by the property of involving the parameters
yx, ..., yn linearly in the finite equations of the group.
We may suppose that the coordinates have been so chosen
that (1, 0, 0, ...) is a point of general position, and therefore,
the group being transitive, we may transform this point to
any arbitrarily selected point by a transformation of the
group ; it is therefore necessary that the n linear functions
I- = n
2a^*> (s = l,...,n)
should be independent.
If we now introduce a new set of parameters zl,.,.,zn
defined by s. = 2«.<*9*.
the equations of the group will take the form
(2) *i =2 &*«<**;
and, since the coefficient of xx must be zs, we shall have
(3) /3slA; = e«fc»
where esli is equal to unity if s = k, and to zero otherwise.
* In this chapter I have made much use of §§ 3, 4 in Chapter XXI of
Lie-Scheffers' Vorlesungen iiber continuierliche Gruppen.
307] SOME LINEAR GROUPS 399
The equations (2) define a group which will, we assume,
contain the identical transformation. It must, therefore, be
possible to find zls ..., zn to satisfy the equations
k = n
2i Psik Zh ~ e«i i
and in particular, taking i to be unity, to satisfy the equations
k = n
2* esk zk ~ esi '
so that zx = 1, z2 = 0, ..., zn = 0, and /3gil = esi.
Expressing the fact that the operation, resulting from first
carrying out the operation with the parameters %,..., zn, and
then that with the parameters z^, ..., z'n, must be the same as
the operation with some parameters s£ , ..., z^, we have
i =j = k = I = n k — i = n
(4) 2 PsikPijlxjziz'k=^ PsihZ'k xi> (s= h-^n).
Equating the coeflicient of xY on each side we see by (3) that
(5) z's' = 2 P,ik Psil zi z'k = 2 Psik zi zk •
These equations give the parameters z[' , . . . , z'^ ; and if we
substitute their values on the right of the equation (4), and
then equate the coefficients of the variables on each side we
obtain, as the necessary and sufficient conditions (in addition
to j38lj. = /3g&1 = csk) in order that (2) may be the equations
of a group
i = n i = «
(6) 2 Psik Pijz = 2 Psji fiilk
for all values of s, k, j, I from 1 to n inclusive.
A linear group of the form (1) when thrown into the form
(2) is said to be in standard form ; from (5) we see that the
group in standard form is its own parameter group.
By interchanging k and j in (2) we see that the equations
k = i = n
(7) fl£ =2 &*<»«**« (8=1, ...,»)
also define a linear group in standard form, and with the
parameters only involved linearly.
The condition that the linear transformations
j = n j=n
lij Xj
x'i = 2 aa xj and xi = 2 hij xi
400 LINEAR GROUPS INVOLVING [307
may be permutable is
;' = n j — n
(8) 2«y6jft=26y-«
we therefore see from (6) that every operation of (2) is per-
mutable with every operation of (7) ; the two groups are
then reciprocal.
§ 308. Conversely, any simply transitive linear group, whose
reciprocal group is also linear, must be of the form (2) of
§ 307. We prove this as follows :
If Sx , . . . , Sr are a number of linear- transformations (which
need not form a group), we say that the linear transformation
A.j Oj + . . . + Ar Sr ,
where A15 ..., Ar are constants, is dependent on S1} ..., Sr.
It is clear that in n variables there cannot be more than
n2 independent linear transformations.
If we are given r linear transformations Sv ...,#,. we cannot
in general find a linear transformation T permutable with
each of them ; the forms of the given transformations, however,
may be such that there are a number of linear transformations
permutable with them.
Let Tlt ..., Tg be the totality of all independent linear
transformations permutable with 8lt ..., Sr. The condition
that two linear transformations should be permutable shows
us that every linear transformation dependent on Tx, ..., Ts is
permutable with every linear transformation dependent on
S1} ..., Sr. Now Ti Tj is linear and permutable with S±, ..., Sr;
it must therefore be dependent upon T1,...,T8, and therefore,
from first principles, 2\, ..., Ts form a finite continuous group
into which the parameters enter linearly.
The operations 81} ...,Sr must now be operations of a linear
group of the class we are now considering. For SjSj is a
linear transformation, permutable with Tlt ...,TS; and there-
fore from S1,..., Sr we can generate a group which will be
linear, permutable with Tlt ..., Ts, and will include amongst
its operations Sls...3'Sr.
The two groups 813 S2i ... and Tx, T2, ... will be permutable
and each will involve the parameters linearly.
Let Sx, ..., Sn be a simply transitive linear group G, with
the special property that its reciprocal group T (which is of
course simply transitive) is also linear in the variables. By
what we have proved T must involve the parameters linearly;
310] THE PARAMETERS LINEARLY 401
and therefore G being the reciprocal group of T must do
likewise ; and therefore be of the form (7) of § 307.
§ 309. The linear operators of (2) § 307 are given by
i = i = n
Xk =2 P$a xiTZ> (& = 1, ..., n),
0J-s
and in particular the group contains
Zi=2^t->
VJ'8
which is permutable with every other linear operator.
A linear group therefore in which the parameters enter
linearly must always contain the Abelian operator
i = n
If we are given the infinitesimal operators of a simply
transitive linear group we may at once determine whether or
not it belongs to the class of groups we are here considering.
Let these operators be
$ = i = n
Xh=^aeik:/:i^7> (k=l,...,n);
CU/8
then, if the group is of the required class, we know that the
finite transformations must be given by
t = Jfc = n
^=2 asikxiyk>
and therefore if, and only if, these equations generate a group,
will the given group be of the required class.
§ 310. We shall now determine all possible groups of this
class in three variables.
First we shall prove that the alternant of two linear opera-
tors can never be equal to the linear operator
£7=2
x
g1>x
8
The operators of the general linear homogeneous group are
X{ - — . ..., where i and k are any integers from 1 to n ;
CAMPBELL J) d
402 LINEAR GROUPS INVOLVING [310
and the operators of the special linear group are ^- — , ...,
where i and h are unequal, and also xi - xJc r— •
o X^ a Xfc
This operator U cannot then belong to the special linear
group ; the alternant therefore of two operators of the special
linear group can never be equal to U.
Now if Ar is any linear operator whatever, we can find
a constant A making X + A U an operator of the special linear
group. We then take (X and Y being any two linear opera-
tors) X + \U and Y+nU to be two operators of this special
group. We have to prove that (X, Y) cannot be equal to U ; if
it were equal to U then (X + \U, Y+fxU), being identically
equal to (X, Y), would be equal to U ; and we have just
proved that this is impossible.
Let now X, Y, U be the operators of a group of the re-
quired class, viz. one in which the parameters enter the finite
equations linearly. The operator U being permutable with
every linear operator, we have
(U, X) = 0, (U,Y) = 0, (X,Y) = aX + bY + cU,
where a, b, c are some constants. We have just proved that
a and b cannot both be zero unless c is zero ; if a, b, c are all
zero the group has the structure
(1) (U,X) = 0, (U,Y) = o, (X, Y) = 0.
Now this group is Abelian, and therefore, if linear, must
be of the required class ; for its reciprocal group coincides
with it, and is therefore linear, and by § 308 must therefore
involve the parameters linearly in its finite equations.
If a and b are not both zero, and we take operators of
the form X + kU, Y+ixlI, and U as fundamental operators
of the group, we can cause c to disappear from the structure
constants ; and we then see that fundamental operators may
be so chosen that the group will have the structure
(2) (U,X) = o, (U,Y) = o, (X,Y) = X.
From what we have proved in § 263, we see that any linear
operator in the variables x, y, z must be of one of the
following types :
xp + byq +cU, where b and c are constants and b^l ;
(3) xp + ezq +cU, where e is zero or unity;
e^p + e.-^zq + cU, where ex and e2 are unity or zero.
310] THE PARAMETERS LINEARLY 403
We therefore can take X to be of one of the following types
(since the group has U as one of its operators) :
(4) xp + byq, where b is neither zero nor unity ;
(5) xp; (6) xp + zq; (7) yp + zq; (8) zq.
We must then find F from the identity IX, Y) = 0, or
from (X, F) = X.
Let the third operator of the group be
ox oy dz
where £, rj, ( are linear and homogeneous functions which can
be found from the structure constants when we know X; in
finding Y we may omit any part which is dependent on
X and U.
Take X in the form (4) and form its alternant with F;
we have
x£i + ty£2-£= A£ x^ + by^-brj^Xby, xCx + by(2=0,
where k is zero if the group is in Class (l) and unity if in
Class (2) ; we then find that the only possible group is in the
first class and is
(A) xp, yq, zr.
Taking X in the form (5), we see that the group must
contain yq + zr; and, if it is in Class (l), F must be of the form
fay + a2z) q + fay + a±z) r.
Omitting the part yq + zr we can reduce this, by § 263,
to one of the two forms yq — zr or zq; the group is therefore
either of the form
(B) xp, zq, xp + yq + zr,
or it is of the form (A).
It may be shown that there is no group in Class (2) with X
in the form (5).
It may also be verified that (6) does not lead to a new
group.
Passing to (7), we see that in Class (l) F must be of the
form zp; if Fis in Class (2) it may be reduced to the form
xp—zr by a linear transformation.
D d i
404 LINEAR GROUPS INVOLVING [310
We therefore have the two groups
(C) VP + zq, zp, xp + yq + zr;
(D) yp + zq, xp — zr, xp + yq + zr.
We next take X to be zq ; if the group is in Class (1), we
have -t-r , x
Y = [axx + a2z) p + a3xq.
We cannot have ax = a2 = 0, for this would make the group
intransitive.
If ax = 0 but neither a2 nor a3 is equal to zero, we have
the type (C) again.
If ax = a3 = 0 we get the type
(E) zp, zq, xp+yq + zr.
If ax is not zero, we may reduce (by linear transformation)
Y to the form axxp; we thus obtain the type (B) again.
If the group is in Class (2) and X = zq, we have
Y= (axx + a2z) p + (y + a3x) q.
If ax = 0, then, the group being transitive, a., cannot be
zero ; by a transformation of the form
x'—x + vz, y'=y + \x, z' = z,
we may then reduce Y to the form yq + zp.
This gives the group
(F) zq, yq + zp, xp + yq + zr.
If ax = 1, we may so transform that
Y= a3xq—zr;
if a3 is not zero, this gives the group
(G) zq, xq + zr, xp + yq + zr;
if a3 is zero, we have the group
(H) zq, xp + yq, zr.
If ax is neither zero nor unity, we may reduce F to the form
axp + yq ;
and we then have the group
(I) zq, axp + yq, xp + yq + zr,
where a is neither zero nor unity.
311] THE PARAMETERS LINEARLY 405
§ 311. We must now examine all these groups to see
whether the parameters occur linearly in the finite equations
of the groups.
The finite equations corresponding to (A) are
*'= exx, y'= e2y, z'= e3z.
The point (1, 0, 0) is not, however, a point of general
position, since the coefficients of x in the three equations are
not independent linear functions of the parameters.
These equations clearly form a group with the property of
being its own parameter group. The group is not, however,
in what we have defined as standard form, though it can be
brought to that form. To bring it to standard form it is
necessary to transform it so that in the new coordinates the
point (1, 0, 0) may be one of general position. We therefore
take
X-* — X) X.) — X ~r 2/, X3 — X "i~ Zt
2/l == ei> Vl = ei~e2> 2/3 = ei~e3»
and thus obtain the group
(A) x\ = y1x1, d2 = y2xx + (yx-y2)x2, *'3 = 2/3*1 + (2/i -2/3K •
This group is one of the class required and is in standard form.
The finite equations which correspond to (B) are
x'= (ex + e3) x, £/'= e3y + e2z, z'= e3z.
If we take x1 = 0, x2 = y, x3 =» x+ z,
2/i = ~ez"> 2/2 = e2» 2/3 = eu
we have a group of the required class
(B) x\ = yxxx, 4 = 2/2*i + 2/i*2» 4 = 2/3*1 + (2/i -2/3) «V
The operators (C) lead to the group
(C) x[ = y1xli x'2=y2xx + yxx„ xr3=y3x1 + y2x2 + y1x.i,
which is of the required class and in standard form.
If the operators (D) lead to a group whose finite equations
involve the parameters linearly, the equations in finite form
must be
x' = (e2 + e3) x + exy, y'=e3y + exz, z' = (e3 - e2) z.
Now these are not the equations of a group at all, so that
the equations (D) do not lead to a group of the type we want.
Similarly we see that (F), (G), and (I) do not lead to the
required type of group.
406 THE THEORY OF [311
The operators (E) lead to
(E) x[ = yxxx , x2 = y2xx + yxx2, x'z = y3xx + yx x3.
Finally the operators (H) lead to
(H) af^y^, x2=y2xx + yxx2 + y2x3, x3=y3xx + (yx + y3)x3.
There are, therefore, only five types of groups in three
variables which are linear in both variables and parameters ;
and of these groups only (H) is non-Abelian.
An example of a non-Abelian group linear in four variables
and four parameters is
xi = y\x\~Vix'%"^V%x,i~VvJC\i
aj2== 2/2 Xl + Vl X2 ~ 2/4 ^3 ~~ 2/3^4'
X3 — VzX\ ~y±X2 + V\XZ + 2/2^4 >
a'4= 2/4^1 ^VzX2~ 2/2^3"^ 2/l Xi'
An example of an Abelian linear group in five variables is
xx =yxxx,
X2 ~ 2/2*^1 ' 2/l X2 '
X3 = 2/3 Xl "■" 2/2^2 + 2/l^3 '
x\ = y±xx + 2/3 x2 + y2x3 + yxxA,
X'5 = 2/5^1 + 2/4^2 + 2/3^3 + 2/2^4 + 2/l *5*-
§ 312. We now proceed to explain the connexion of these
results with the theory of higher complex numbers.
Let ex, ..., en be a system of n independent complex num-
bers ; any number x of the system can be expressed in the
form
X — Xx € x -J- . . , -+- Xft 6yi ,
where xx, ..., xn are ordinary numbers; x can therefore only
be equal to zero when xx, ...,xn are each zero.
We call ex, ..., en the fundamental complex numbers of the
system ; but if /3X , . . . , f3n are any n independent complex
numbers of the system we could equally take them to be
the fundamental complex numbers, and express all other
numbers in terms of them.
From the fact that the number resulting from the multi-
plication of two complex numbers must be expressible in
terms of the fundamental complex numbers we have
* Burnside, Proceedings of the London Mathematical Society, XXIX, p. 339.
312] HIGHER COMPLEX NUMBERS 407
where yjki, ... are a system of ordinary numbers, fixed when
we have chosen our fundamental complex numbers. If,
therefore, u is the complex number yx,
i = k = n
u8=^ySkiykxi'
Similarly, if v is the complex number xy,
i = k = n
vs=^ySikykxi-
From the fact that division is to be an operation possible
in the system — that is, when we are given x and u, or x and
v, we must be able in general to determine y — we see that
the determinant Mx whose sth row and kth column is
i = n
j^ 7$kixii
cannot vanish identically ; nor can the determinant M'x, whose
i = n
sth row and kth column is 2 ysikxi> vanish identically.
It follows, therefore, that the equation system
t = k = n
0) <=2 ysuVk^ (s = !> •••> n)>
where we look on xv ..., xn as the original variables, and
#'i) ••■>xn as the transformed, is such that the determinant
of the transformation does not vanish.
For a similar reason the determinant of
t = k = n
(2) x's=^ysikVkxi
does not vanish.
Since in the system of complex numbers the law of multi-
plication is to be associative, if u = yx and v = zy, we must
have zu — vx. Therefore
t = i = n t = k = n
2 zi ut ysit es = 2 ^ xk ystk es ; and therefore
t = i=j = k = n t = i =j = k = n
2 zi es ySit vtjk Vj xk=^xk ystk es yuj zi Vj •
Equating the coefficients of z^ e8 xk y? on each side we have
408 THE GROUPS CORRESPONDING TO [312
t = n t=n
(3) 2 7 sit YtjTc = 2 Vstk Vtij •
Now these are just the conditions that (1) should generate
a group which is its own parameter group, and they are equally
the conditions that (2) should do so.
§ 313. We must now prove that these groups contain the
identical transformation.
Let x = x1e1+ ... + xnen be a general complex number, that
is, a number such that neither Mx nor Mx is zero ; we can,
whatever u may be, find a complex number y such that u
is equal to yx. Now let u be taken equal to x, and let the
corresponding number y be denoted by e, so that x is equal
to e x ; we shall prove that e does not depend on x at all, and
shall investigate its position in the system.
Let v be any other general complex number, and z a com-
plex such that v is equal to xz ; we have
ev = €xz = xz = v ;
that is, e has the same relation to v as to x, and therefore does
not depend on either v or x.
Next we see that if yx is zero, where # is a general complex
number, we must have, since Mx is not zero,
2/i = °> •••> yn= °-
So, since M'x is not zero, if xy is zero, we must have
2/i= °>--.> Vn= °-
Let x' be equal to xe, then
and therefore {xf —x)x is zero, so that x' is equal to x\ that
is, we also have x = xe.
This unique number e is therefore a complex unity.
Let « = e1e1 + ... + enen, where e1} ..., en are ordinary num-
bers, then, since x = xe = ex, we have
^ = 2 Vsik xi(k=^ ysu €k xi ■
We now see that yk = ek, (k=l,...,n)
will give the identical transformation in (1) and (2) of § 312.
The two equation systems, therefore, define groups each
containing the identical transformation ; and, since neither
315] SYSTEMS OF COMPLEX NUMBERS 409
Mx nor M'x is zero, there are n effective parameters ; that is,
the groups are simply transitive, and involve the parameters
linearly, and each group has the property of being its own
parameter group.
If we were to take e as one of our fundamental complex
numbers, say e1} we should have each group in its standard
form.
§ 314. The infinitesimal operators of (1), §31 2, are Xlt ...,Xn,
i = s = n
where Xh =^7sikxi^-
and (*#.*i)=2*¥*/-
S=t=j=n ^
Now (X{ ,Xk)=2, (ySji Ytsk - y$jk Ytai) xj ^ »
s = t =j = n
=2 (ysik-Yski) nj*xj^' h7 (3) of § 312>
S = 11
=^(ysik-y8M)X8>
and therefore ciks - ysik -ysM .
Similarly we may write down the operators of the group
(2) of § 312 ; and it may be at once verified (by aid of (3)
§ 312) that the two sets of operators are permutable, so that
the groups are reciprocal.
We thus see that to every system of complex numbers there
will correspond two simply transitive reciprocal linear groups ;
and conversely, to every pair of such groups a system of com-
plex numbers.
The complex number e whose existence we have proved may
be taken to be an ordinary unit number since ex = xc = x.
The fundamental complex numbers may therefore be taken
to be the ordinary unity and e2, ..., en as in the Hamiltonian
Quaternion system.
§ 315. When we are given a simply transitive linear group
in standard form, and wish to write down the corresponding
system of complex numbers, we multiply x\ by elt x'2 by e2, ...
and, adding, equate the coefficient of x^^, on the right of the
transformation scheme, to ct-e^.
The laws of combination of the symbols elt ..., en are most
conveniently expressed in the form of a square of n2 com-
410
SYSTEMS OF COMPLEX NUMBERS
[315
partments, the expression equal to e^e^ being found in the
compartment corresponding to the ith row and kth column.
Thus the system corresponding to (H) is denoted by
this means that
e.
el
H
ez
H
0
0
ez
0
63
ei — ^u ^2 — ^' 63 — ^3> ^1^2 — ^2' ^2^1 — ^2>
^lg3 == 63' e3gl = 63' ^63 = ^J e3^2 == 62 >
where we understand that the operation on the right in e^ ek is
to be taken first.
The other systems in three complex numbers are all com-
mutative, since the groups are Abelian.
The non- Abelian group of order four gives the system
&i 6o €"> €■»
el
e2
e3
ei
e2
~ei
ei
~h
63
-«4
~ei
e-z
H
%
~e2
~«1
i. e. the Hamiltonian Quaternion system, when we take ex = 1 .
INDEX
The numbers refer to the pages.
Abelian group, definition of, 17 ;
simplest form of, when all its
operators are unconnected, 85.
Abelian operations of a group,
definition of, 16; condition that
a group may have, 71 ; if a group
has none, it has the structure of
the linear group (the adjoint
group) given, 73.
Abelian sub-system of functions,
definition of, 218.
Admit, when an operation is said
to be admitted by a group, 16 ;
an infinitesimal transformation,
by a function, 82 ; by a complete
system of operators or of differen-
tial equations, 93 ; a contact
transformation by a function
or equation, 278.
Alternant, of two linear operators,
definition of, 8 ; of two functions,
196.
Ampere's equation, when it can
be transformed to s=0, 243;
the group then admitted, 307.
Bilinear equations, defining a
contact transformation, 257 ;
simplified by projective trans-
formation, 257, 268.
Burnside, quoted, 2, 165, 406.
Canonical equations of a group,
45 ; relation between canonical
parameters of an operation and
its inverse, 46 ; canonical form
varies with choice of fundamen-
tal operators, 162.
Characteristic function of an
infinitesimal contact transforma-
tion, 277 ; of the alternant of
TFj and W2, 285 ; of the contact
operator of the plane x, z re-
garded as an operator in space
x, y, z, 371.
Characteristic manifold of an
equation or function, defini-
tion of, 279 ; properties of, 279,
280 ; one passes through every
element of space, 279.
Co-gredient transformation
schemes, definition of, 15.
Complete system of homoge-
neous functions, definition of,
213, 215 ; if of degree zero, in
involution, 215; reduced to sim-
plest form, 222, 223 ; is a sub-
system within a system not con-
taining Abelian functions, 224 ;
can be transformed by a homo-
geneous contact transformation
to any other system of the same
structure, 235.
Complete system of linear par-
tial differential equations, con-
dition that they should admit an
infinitesimal transformation, 93.
Complete system of operators,
definition of, 82 ; in normal form,
83 ; when permutable, 84.
Complex numbers, connexion of,
with a class of linear groups,
406-410.
Complexes, linear, of lines, ele-
mentary properties of, 255-257 ;
tetrahedral, 269.
Conformal group, 32 ; isomorphic
with the projective group of a
linear complex, 305, 396.
Conjugate elements, definition of,
260.
Conjugate operations, definition
of, 16.
Conjugate sub-group, definition
of, 17 ; method of finding all,
183-185.
412
INDEX
Contact groups, fundamental
theorems on, 287-290 ; when
similar, 290 ; when reducible,
292 ; connexion with Pfaff's
Problem, 293 ; in the plane re-
garded as point groups in space,
302.
Contact transformations, homo-
geneous, definition of, 228; given
when Xv ..., Xn given, 229;
when one set of functions can be
transformed to another by aid
of, 236 ; infinitesimal, 276. See
also under Extended.
Contact transformations, non-
homogeneous, definition of, 240 ;
generate a group, 241 ; infini-
tesimal, 276 ; geometrical inter-
pretation, 280 ; how the infini-
tesimal operator is transformed,
286.
Contact transformation which
transforms straight lines into
spheres, 262 ; points into mini-
mum lines, 261 ; positive and
negative correspondents to a
sphere, 263 ; spheres in contact,
264.
Contact transformation with
symmetrical generating equa-
tions, 268 ; transforms points to
lines of tetrahedral complex,
269 ; planes, to twisted cubics,
269 ; straight lines, to quadrics,
271 ; examples on this method
of transformation, 274.
Continuous group, definition of, 3.
Contracted operators of a group
with respect to equations admit-
ting the group, 128; generate
a group, 129 ; number of uncon-
nected ope rators in thisgroup, 1 30.
Coordinates of a surface, defini-
tion of, 135.
Correspondence established be-
tween the points of two spaces,
151, 152; of isomorphic groups,
162, 163; between manifolds in
two spaces, 262, 268, 304.
Correspondents, positive and
negative, of a sphere, definition
of, 263.
Dependent, when an operator is
said to be, on others, 7.
Differential equation
transformations admitted by, 28.
Differential equation, of the
conic given by the general Car-
tesian equation, 324 ; of the
cuspidal cubic, 326.
Differential equations, partial of
first order, theory of the solution
of linear, admitting known in-
finitesimal transformations, 90-
112; method of finding the
complete integral of non-linear,
204.
Differential invariants of a group
defined, 320 ; how obtained, 320 ;
of the group x'=x, y'= — — ,t
cy + d
321 ; of the projective group of
the plane, 324 ; absolute, 324 ;
of the group of movements in
non-Euclidean space, 330.
Distinct, when infinitesimal trans-
formations are said to be, 95.
Dupin's cyclide, transformed into
a quadric, 265.
Effective parameters, definition
of, 7.
Element, of space, and united
elements, definitions of, 194 ;
linear element, definition of, 280.
Elliott, quoted, 55.
Engel'a theorem, 36.
Equations admitting a given
group, how to obtain, 130;
examples on method, 132.
Equivalent, when two function or
equation systems are said to be,
197.
Euler's transformation for-
mulae, 20.
Extended contact transforma-
tions, operators of, 295 ; in
explicit form for the plane, 296 ;
transforming straight lines to
straight lines, 297 ; circles into
circles, 300 ; transformation of
this group, 302-304 ; explicit
form of operators in space, 305.
Extended operators of, the group
x> = x y'^WlJt 321, 322; the
cy + d
INDEX
413
projective group of the plane,
322, 323 ; the group of move-
ments in non-Euclidean space,
327.
Extended point transformations,
explained, 24 ; formulae for, 24;
illustrative example, 25 ; ex-
tended point group, 288 ; struc-
ture of, 290 ; transforming
straight lines to straight lines,
297 ; circles to circles, 298.
Finite continuous transforma-
tion groups, definition of, 5 ;
origin of theory of, 100 ; contact
groups, 287.
Finite operations of a group
generated from infinitesimal
ones, 45 ; method of obtaining,
47 ; example on method, 48.
Forsyth, quoted, 36, 77, 88, 211,
217.
Fundamental functions used in
invariant theory of groups, 119 ;
how found, 121.
Fundamental theorems on
groups, first, 38, and its con-
verse, 66 ; second, 51, and con-
verse, 57-59 ; third, 68, converse,
75 ; resume, 80 ; similar theo-
rems hold for contact groups,
287-290.
Generating equations of a Pfaf-
fian system, definition of, 196 ;
of a contact transformation, defi-
nition of, 245 ; property of, 246 ;
limitations on, 246 ; interpreta-
tion of limitation, 247 ; applica-
tions of, 252, 259, 268.
Generators of a quadric are
divided in a constant anharmonic
ratio bv any inscribed tetrahe-
dron, 272.
Goursat, quoted, 244.
Group of a point, definition of,
140 ; group locus, definition of,
141 ; stationary and non-sta-
tionary groups, 141 ; when the
point is the origin, 332.
Group of movements in non-
Euclidean space, 327, 395.
Group of movements of a rigid
body in a plane, 18 ; of a net on
a surface, 317.
Group of transformations, gene-
ral definition of, 2 ; continuous,
3, example, 4 ; infinite, 3, ex-
ample, 4; discontinuous, 3, ex-
ample, 4 ; mixed group, 3 ; finite
and continuous, 5, example, 6.
Groups, in cogredient sets of
variables, 115.
Groups of the linear complex,
304, 388.
Groups, possible types of, in a
single variable, 335.
Hamiltonian Quaternion system,
410.
Homogeneous function systems,
defined, 198 ; equation systems,
198 ; condition that a system
should be homogeneous, 214.
See also under Complete.
Identical transformation, defini-
tion of, 3 ; parameters defining,
Imprimitive groups, definition
of, 137 ; admitted by a complete
system, 139 ; of the plane, di-
vided into four classes, 353 ; all
types of these groups found,
354-364; arranged into mutually
exclusive types, 368.
Independent, infinitesimal trans-
formations, 7 ; linear operators,
7 ; functions, 81.
Index of sub-group, definition of,
183.
Infinitesimal transformation,
definition of, 6 ; operator, defini-
tion of, 6 ; operators of first
parameter group, 41 ; are un-
connected, 45.
Integral cones, elementary, defini-
tion of, 281 ; associated differen-
tial equation, 282.
Integral of a differential equation,
Lie's extension of definition,
202, 231, 232.
Integration operations, definition
of, 88.
Invariant curve systems of the
imprimitive groups of the plane,
366, 367.
Invariants, of a complete system
of operators, 87 ; transformed to
other invariants by any trans-
414
INDEX
formation which the system
admits, 94 ; of an intransitive
group, 114; geometrical inter-
pretation, 114.
Invariant. See under Differential.
Invariant, theory of binary quan-
tics, 118; equations with respect
to a group, 128 ; how obtained,
130; decomposition of space, 137.
Inverse transformation scheme,
1.
Involution, functions in, defini-
tion of, 197 ; equations in, 197 ;
if any equation system is in
involution, so is any equivalent
system, 197; contact transforma-
tion admitted by equation system
in, 278.
Irreducible contact groups of
the plane obtained, 371-378;
types of, enumerated, 378, 380.
Isomorphic, two groups are simply
isomorphic when they have the
same parameter group, 162.
Isomorphism of two groups,
simple, definition of, 10 ; ex-
ample of, 10 ; multiple, defini-
tion of, 163 ; when a group is
multiply isomorphic with an-
other, a self-conjugate sub-group
in the first corresponds to the
identical transformation in the
second, 164.
Jacobian identity, definition of,
67; identity deduced from, 216.
Linear complex, definition of,
255 ; form to which it can be
reduced, 256; lines conjugate
with respect to, 256 ; complexes
in involution, 257 ; projective
group of, 304.
Linear groups whose finite equa-
tions involve the parameters
linearly, 398-401; standard form
of such a group, 399 ; must
contain an Abelian operator,
401 ; enumeration of such groups
in three variables, 405,406 ; con-
nexion with the theory of higher
complex numbers, 406-410.
Linear homogeneous group,
general, 14, special, 17 ; simpli-
fication of the form of an operator
of, 336-338 ; possible types ot,
in two variables, 339, 341.
Linear operators, any one is
of type — > 84 ; transformation
formula for any operator. 91 ;
formal laws of combination of,
54-57.
Lines of curvature transformed
to lines of inflection, 266.
Manifolds of united elements,
definition of, 201 ; the symbol
!/„_! 201 ; different classes of,
201;' in ordinary 3-way space,
250.
Maximum sub-group, definition
of, 101.
Measure of curvature unaltered
by transformations which do not
alter length of arc, 310 ; expres-
sion for, 315 ; constant along
lines of motion of points of a net,
312.
Minimum curves, definition of,
28.
Mongian equations, defined, 29 ;
associated with an equation of
first order, 28, 282; of tetra-
hedral complex, 282.
Non-homogeneous contact trans-
formation, 240.
Non-stationary group, defined,
141.
Normal form of complete system
of operators, 83 ; operators are
permutable, 84.
Normal structure constants,
defined, 72.
Null plane, definition of, 256.
Operators of a group, definition
of, 37 ; fundamental theorem on,
38 ; number of independent, 38 ;
examples on finding, 40, 41 ;
condition that one may be self-
conjugate, Abelian, 93 ; arranged
in classes according to their
degrees in the variables, 332.
Order of a group, definition of,
18 ; of an integration operation,
88 ; of a Pfaffian system of equa-
tions, 196.
INDEX
415
Parameter group, first and second,
definitions of, 13 ; any operation
of the first permutable with any
operation of the second, 13 ;
parameter groups of general
linear homogeneous group, 15 ;
structure constants of, 65, 159 ;
operators of, 160, 161 ; of two
simply isomorphic groups iden-
tical, 162.
Permutable operations, definition
of, 2 ; condition that two linear
transformations may be, 400.
Pfaffian system, definition of,
196 ; condition that given system
of equations should form, 201 ;
transformation of, 231.
PfafFs equation, definition of,
194 ; solution, 195 ; in non-
homogeneous form, 238.
Pfaff's problem, in relation to
contact transformation, 293.
Poineare, quoted, 36.
Polar system of functions to a
given complete system, 217 ; if
given system is homogeneous,
polar is also, 217.
Primitive groups, definition of,
137; possible types of, in the
plane, 352 ; in space, 397.
Projective groups and sub-groups,
18, 20 ; examples of non-projec-
tive groups, 19, 22 ; of the linear
complex, 304, 388 ; of the plane,
property of sub-group of, 385.
Reciprocal groups, definition of,
62 ; structure constants of, 158.
Reciprocation, a case of contact
transformation, 252.
Reduced operators, definition of,
97.
Reducible contact groups; 292 ;
of the plane, condition for, 370.
Salmon, quoted, 265, 266, 315.
Scheffers, quoted, 272, 398.
Self-conjugate operator, condi-
tion for, 93.
Self- conjugate sub-group, defi-
nition of, 17 ; condition that a
given sub-group may be, 92.
Similar groups, definition of, 16 ;
are simply isomorphic, 16;
necessary and sufficient con-
ditions that two groups may
be similar, 149-154 ; that two
contact groups may be, 290, 291.
Similar operations, definition of, 2.
Simple group, definition of, 165.
Special elements, definition of,
249 ; equations satisfied by, 249,
254.
Special envelope, definition of,
249.
Special equations, definition of,
247.
Special linear homogeneous
group, definition of, 17.
Special position, points of, with
respect to a complete system of
operators, 110; transformed to
points of the same special order
by transformations admitted by
system, 127.
Standard form of a group, defini-
tion of, 147 ; of a homogeneous
function system, 198.
Stationary functions, definition
of, 144 ; construction of, 187.
Stationary group, definition of,
141; all such groups imprimitive,
142 ; operators permutable with,
156, 157.
Structure, when two groups are
said to be of the same, 70.
Structure constants, definition of
a set of, 68; vary with choice
of fundamental operators, 70 ;
normal structure constants, 72 ;
a set resulting from a change
of fundamental operators, 177 ;
construction of group, when
structure constants given, 187 ;
examples on, 189-192 ; structure
constants of contact group, 292.
Structure functions of a complete
system of operators, definition
of, 144 ; of a complete system of
functions, 215.
Sub-group, definition of, 17 ;
maximum, 101 ; equations de-
fining a, 181; index of, 183;
method of finding all types of,
186 ; examples on method, 189-
192.
Surface coordinates, 313, 314.
Surfaces on which a net can move,
311-318 ; group of movements of
416
INDEX
the net, 317 ; when the surface
is a developable, 318.
Tetrahedral complex, definition
of, 269 ; Mongian equation satis-
fied by linear elements of, 282.
Transformation group, general
definition of, 2.
Transformations which transform
surfaces but leave unaltered
length of arcs, 308-311.
Transitive group, simply transi-
tive group, definitions of, 45, 113 ;
when two transitive groups are
similar, 167 ; construction of,
when the structure constants and
stationary functions are given,
170-173 ; extension to the case
of intransitive groups, 174.
Translation group, 18.
Trivial, when infinitesimal trans-
formations admitted by an equa-
tion are said to be, 95.
Type, when groups are said to be
of the same, 16 ; when sub-
groups, 17 ; number of types of
groups, 22.
Unconnected, operators, defined,
7 ; functions, 81 ; infinitesimal
transformations, 82 ; invariants
of a complete system, 83.
United elements, definition of,
194.
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