NA TURE 



49 



THURSDAY, NOVEMBER 17, 1904- 



THE THEORY OF CONTINUOUS GROUPS. 



Introductory Treatise on Lie's Theory of Finite Con- 

 tinuous Transformation Groups. By John Edward 

 Campbell, M.A., Fellow and Tutor of Hertford 

 College, Oxford, and Mathematical Lecturer at 

 University College, Oxford. Pp. xx + 416. (Oxford: 

 Clarendon Press, 1903.) Price 14s. net. 



THE theory of continuous groups should appeal to 

 all who are interested in mathematics ; it is based 

 on the fundamental ideas involved in cases of change 

 of the algebraic notation, and as such is an illumin- 

 ating synthesis of a large number of our elementary 

 operations; and the principal notions of the theory, 

 once laid bare, are so simple and admit of so many 

 familiar applications that these should form an 

 integral part of elementary teaching, particularly in 

 analytical geometry and differential equations. As to 

 its philosophical import, the theory is of the greatest 

 value in the analysis of our geometrical conceptions, 

 being an indispensable part of that algebraic scheme 

 which, at present running parallel with these, may 

 modify them still more than hitherto before the 

 parallelism is recognised again as an identity. 



Lie himself, though directing attention to the fact 

 that he heard as a student, in 1863, lectures from 

 Sylow on Galois's theory of discontinuous groups, and 

 acknowledging his indebtedness to several writers on 

 partial differential equations, would seem to have been 

 interested, above all other things, in the transform- 

 ations of analytical geometry; and while the precise 

 propositions of his theory of groups must be primarily 

 attributed to his study of systems of linear partial 

 differential equations, his bias was at first, and largely 

 throughout, to arrive at his conclusions by the help 

 of geometrical intuition. Thus, though he has 

 succeeded so extraordinarily in what he tells us was 

 one of his objects, drawing again into organic union 

 branches of mathematics which threatened to pursue 

 solitary developments, there is, some may think, a 

 certain underlying vagueness of definition as to the 

 character of the functions to which his theories apply. 

 This even has, perhaps, some advantages. 



Of these various points of view the book now under 

 notice gives the English student an excellent means 

 of judging. With roughly the same purpose as the 

 simplified German account of Lie's theory (Scheffers, 

 1893, 800 pages), it is briefer, and yet quite clear in 

 statement; it contains more of the application of Lie's 

 theory to the solution of partial differential equations, 

 and it offers alternative proofs, due to its writer, of the 

 fundamental theorems of the subject. Like the 

 German book, it largely leaves aside the developments 

 subsequent to Lie, such as the intricate theory of the 

 structure of groups, and the application to the trans- 

 formation group of systems of differential equations 

 initiated by Picard, and leaves wholly aside Lie's 

 criticism of the axioms of geometry, while it accepts 

 'Lie's function theory throughout; but it abounds in 

 apt examples, chosen mainly from differential equa- 

 tions and geometry, so that almost any mathematical 

 NO. 1829, VOL. 71] 



student may find something to interest him, and, with 

 such limitations as noticed above, it is extraordinarily 

 full and complete. Altogether a book which should be 

 widely read. 



So much so that it is both difficult and uncongenial 

 to offer any criticisms, were only a review complete 

 without some. To us it seems that some account of 

 systems of equations w-hich in the aggregate define a 

 finite continuous group forms the most natural intro- 

 duction to the theory; though Lie's account of them 

 comes near the end of his third volume he is there 

 revising his fundamental principles, and the ideas in- 

 volved are very simple. Reference to Schlesinger's 

 " Treatise on Linear Differential Equations " (Bd. ii.. 

 Tell i., p. 23) shows how this suggestion works out 

 in detail. It seems right that the student should early 

 learn, for instance, how far the linear transformations 

 which leave x^ + y- unaltered fall under Lie's termin- 

 ology. Perhaps, again, fuller references to anticipa- 

 tions of the ideas which Lie has coordinated into one 

 system would have helped the student. Such may be 

 found widely scattered in all the early masters ; two 

 that are handy to us are in Sylvester's writings. In 

 1852, when Lie was ten years old, Sylvester (" Collected 

 Works," vol. i., pp. 326, 353), while ascribing the 

 notion partly to others, writes of continuous or 

 infinitesimal variation, and that " concomitance can- 

 not exist for infinitesimal variations without, by 

 necessary implication, existing for finite variations 

 also." Or, again, the deduction, so interesting when 

 we first came across it, of the equations for the in- 

 finitesimal motion of a rigid body, from the invariance 

 of the expression dx' + dy' + dz', is in a paper of 

 Sylvester's of 1839 {ib., p. 34). Again, it 

 appears to us, though recognising the value of Mr. 

 Campbell's proofs of the fundamental theorems, that 

 much would have been gained in directness, with- 

 out appreciable increase of the necessarily analytical 

 character of much of the subject, by a frank recognition 

 of Schur's forms for the first parameter group in terms 

 of the constants of structure; of this we are, perhaps, 

 not impartial judges (see Proc. Lond. Math. Soc, 

 vol. xxxiv. p. 91), as equally not of Mr. Campbell's 

 use of the word united in his exposition of Lie's de- 

 finition of an integral of a partial differential equation, 

 having ventured elsewhere to introduce the words 

 connected and connectivity, which latter seems better 

 than the mere symbol M„ which Mr. Campbell adopts 

 from Lie (see " Encyc. Brit.," vol. xxvii. p. 452). But 

 we have a more serious contention with Mr. Campbell 

 about a matter in which opinions will be widely divided ; 

 no doubt it is proper that a beginner's course in the 

 theory of groups should insist primarily on the group 

 property, and not confuse this by complicated consider- 

 ations in regard to the properties of functions ; but in 

 our opinion no account can be regarded as modern 

 which does not face the difficulties ; it seems to us mis- 

 leading, without careful explanations, to use language 

 about functions in general which applies in the first 

 ! instance only to the simplest algebraic functions. On 

 p. II we read : " fej can in general be expressed . . . 

 I in order that (2) may remain an analytic function of 

 its arguments." In what way is the student to 

 imagine the function defined after it has ceased to be 



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