June 2 i, 1900] 



NA TURE 



71 



now been published are so far complete in themselves 

 that it is possible to give some account of their contents, 

 and to appreciate, to some extent, the author's method 

 and point of view. 



Part i. treats of exact equations and the problem of 

 Pfaff. Of the two chapters on exact equations it is 

 enough to say that they contain an excellent summary, 

 with well-chosen examples, of the various methods which 

 have been suggested ; the most interesting part is that 

 which deals with Mayer's very remarkable extensioq of 

 Natani's procedure. 



The rest of vol. i. is devoted to Pfaff's problem. A 

 chapter on the history of the problem is followed by ten 

 others, which give, in the order of their discovery for the 

 most part, the principal results of Pfaff, Jacobi, Natani, 

 Clebsch, Grassmann, Lie and Frobenius. This plan has 

 its advantages, especially for those who wish to become 

 familiar with the literature of the subject ; and mathe- 

 matical experts will duly appreciate the service which 

 Prof. Forsyth has done them. But if we look at the 

 result as a text-book for mathematical students, it is a 

 question whether the course taken is the best one. A 

 chapter which is an excellent guide to a reader who has 

 before him the original book or memoir upon which it is 

 based, may be simply puzzling to a student unfamiliar 

 with the subject, and unable to refer to the primary 

 sources. It is doubtful, for instance, if any one who has 

 not mastered the Ausdehnunoslehre will be able to 

 appreciate the chapter on Grassmann's method : and in 

 the sanie way, the chapters on tangential transformations 

 and Lie's method will not, we fear, do much, in them- 

 selves, to arouse an interest in Lie's magnificent dis- 

 coveries. It is unfortunate that Prof Forsyth's exclusively 

 analytical attitude has prevented him from utilising Lie's 

 geometrical or quasi-geometrical conceptions. It is quite 

 true that intuitional methods require to be controlled by 

 strict analysis ; but they often vivify a mathematical 

 theory in a very instructive and fruitful way. Take, for 

 instance, the question of the "integral equivalent" of 

 the differential relation P</.r-f-QflfK + R^^ = o, where 

 P, Q, R are functions of .x\y, z. If we take x^y, z as 

 ordinary Cartesian co-ordinates, this relation associates 

 with any point A(.r, y, z) a flat pencil of elementary 

 line-elements, concurrent at A, and lying in a definite 

 plane P(^ -.i-)-|-Q(f/ -j)-l-R(f-2-) = o. Thus we may 

 take the " content " of the differential relation to be 

 either a manifold of oo« line-elements, or of oo^ plane- 

 elements. If the given relation is an "exact equation" 

 </</) = o, the integral <p = c gives us a family of oo' sur- 

 faces, each of which contains oo^ line-elements of the 

 content and co^ plane-elements of it. Moreover, every 

 continuous curve made up of line-elements lies (in 

 general) on one of the integral surfaces c/) = f, and the 

 line and plane elements of the surfaces exhaust the 

 corresponding elements of the content. These con- 

 siderations justify us in saying that ^=t, with c an 

 arbitrary constant, is a complete integral equivalent of the 

 differential relation. But in a case \\kexdv + zdy-ydz = o, 

 we cannot construct an integral equivalent of this 

 kind ; and the question arises, what integral equi- 

 valent, if any, exists, and what will be the nature of 

 its equivalence ? To Prof. Forsyth, this is a purely 

 analytical question ; he simply inquires what functional 

 NO. 1599, VOL. 62] 



relations connecting .r,j, 2- are consistent with the given 

 relation. Of the degree and nature of the equivalence 

 to be expected he says very little ; and the gist of what 

 he does say is relegated to a note on p. 250. The geo- 

 metrical theory at once suggests the possibility of con- 

 structing "integral curves" by linking line-elements of 

 the content ; a complete integral equivalent may be con- 

 ceivably constructed by a system of oo^ integral curves 

 together exhausting all the line-elements of the content, 

 or again by ca- integral curves, each with ooi associated 

 plane-elements of the content. As an example of the 

 latter kind of integral equivalent, the system of lines 



X = a, y = bt, z = / 



where a, b are arbitrary constants, and / is a variable 

 parameter, are integral curves derived from the content 

 of av/.r -f- zdy - ydz = o ; and if with each point (a, bt^ t) 

 we associate the elementary flat-pencil which lies in the 

 plane a{x - a) 4- i{y - bz) = o, we have a complete 

 integral equivalent, all the elements of the content being 

 taken into account. If we take the two analytical 

 relations x — a, y = bz, involving arbitrary constants 

 only, we get, it is true, a kind of integral equivalent ; 

 but this is not complete, in any sense analogous to the 

 complete integral of an exact equation. 



Part ii. deals with ordinary equations, not linear ; 

 and the point of view is almost entirely that of function- 

 theory. The coefficients in the equation are analytical 

 functions, in Weierstrass's sense ; and the main problem 

 is that of discussing the functional nature of the depend- 

 ent variable or variables. The discussion is necessarily 

 based upon the work of Cauchy, Briot and Bouquet, 

 Weierstrass and Fuchs ; the analysis is simple enough 

 in essence, but the details, unfortunately, are unavoid- 

 ably lengthy, and tend to be monotonous, owing to the 

 necessity of considering different cases and establishing 

 a set of typical forms. The results are so important 

 that the student is bound to make himself familiar with 

 them ; but the judicious reader will do well to use his 

 privilege of skipping. The fact is, that the demon- 

 strations fall naturally into a very few types ; and it is 

 as profitless to study every one of them minutely as to 

 attempt a detailed examination of every kind of 

 singularity of an algebraic curve. There are, of ccurse, 

 many points in the analysis which cannot fail to arouse 

 interest and admiration ; for instance, the use of a 

 dominant function in proving the existence-theorem, and 

 the employment of a sort of extended Puiseux diagram 

 in the applications. 



Then, again, there are those surprisingly general 

 and definite results which have been deduced, almost as 

 corollaries, from this somewhat unattractive analytical 

 theory. It must suffice to refer to Painlevd's theorem 

 (ii. p. 211), that the points of indeterminateness of every 

 integral of a single equation of a certain very general 

 type are fixed points determined by the differential 

 equation itself ; and to the result established by Bruns 

 (iii. p. 311 and following), that every algebraic integral 

 of the differential equation of the problem of three (or 

 more) bodies can be constructed algebraically from 

 the long-known classical integrals. But the reader 

 will find other results of almost equal interest due to 

 Poincard, Fuchs, Picard and others. The reaction of 



