NATURE 



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THURSDAY, APRIL 4, 1918. 



GOURSAT'S "COURSE OF ANALYSIS/' AND 

 OTHER MATHEMATICAL WORKS. 



(i) A Course in Mathematical Analysis. Differ- 

 ential Equations. Being' part ii. of vol. ii. By 

 Prof. E. Goursat. Translated by Prof. E. R. 

 Hedrick and Otto Dunkel. Pp. viii + 300. 

 (London: Ginn and Co., n.d.) Price 115. 6d. 

 !iet. 



- 1 Finite Collineation Groups, with an Introduc- 

 tion to the Theory of Groups of Operators and 

 Substitution Groups. By Prof. H. F. Blichfeldt. 

 Pp. xi+194. (Chicago, 111.: University of 

 Chicago Press ; London : Cambridge Univer- 

 sity Press, 1917.) Price 1.50 dollars net, or 6s. 

 net. 



3j Introduction to ihe Calculus of Variations. 

 By Prof. W. E. Byerly. Pp. 48. (Mathe- 

 matical Tracts for Physicists.) (Cambridge, 

 Mass. : Harvard University Press ; London : 

 Oxford University Press, 191 7.) Price 35. 6d. 

 ■ net. 



: (i) T^HE American translation of Prof. Gour- 

 J- sat's " Course of Analysis ' ' will be welcome 

 to those who may be unable to read the original 

 easily. The present instalment covers ground in 

 which the author is an acknowledged adept, and 

 it illustrates his remarkable power of illuminating 

 obscurities and giving charm to discussions 

 Avhich, although unavoidable, are apt to be dull. 

 Thus his chapter on existence theorems is not 

 only a model 'of rigour, but actually entertain- 

 ing as well ; § 30, on the Cauchy-Lipschitz 

 method, is most instructive, and illustrates the 

 ^ nlue of a diagram when properly used — not as 

 vehicle for a sham "intuitive proof," but as 

 n image corresponding to a set of analytical 

 data and deductions. Geometrical imagery of this 

 kind is frequently used throughout, and with the 

 happiest results — especially, it seems to us, in the 

 part dealing with partial differential equations of 

 the first order. 



There are some features of special interest in 

 I he earlier part, which deals with ordinary equa- 

 ' >ns. As an isolated gem we may note the inte- 

 ration of Euler's equation (pp. 23-28), especially 

 ihc method which leads to Stieltjes's form of solu- 

 tion. The third part of the chapter on linear 

 ■equations gives a capital summary of the main 

 results obtained by Fuchs, Picard, and others ; 

 illustrations are afforded by the hypergeometric 

 series and Lamp's equation. In the chapter on 

 non-linear ordinary equations of the first order 

 there are a number of valuable results, especially 

 those based on Briot and Bouquet's researches as 

 to equations of the form {dy /dx)^ = R(y), where 

 R(y) is a polynomial in y. Here we have a list 

 of all the cases of this type which can be satisfied 

 ■by a one-valued function of x, and also — which is 

 more important — a clear proof that there are no 

 others. 



, NO. 2527, VOL. lOl] 



Next comes a section on singular solutions, 

 and an Englishman cannot help feeling surprised 

 to find no reference to Cayley here (or, indeed, 

 anywhere else in the volume). Readers should 

 notice the last paragraph of §71; the point is 

 that, if we equate the p-discriminant to zero, the 

 normal meaning of the result is a cusp-locus (or 

 tac-locus, or both) which does not yield a singular 

 solution ; the reason that mathematical students 

 so often obtain a singular solution from the ^-dis- 

 criminant is that so many equations of the type 

 f(x,y, p) = o are made up by eliminating a con- 

 stant c from the equation of a set of algebraic 

 curves (f>{x,y,c) = o, which have an envelope. 



The discussion of Charpit's method seems to 

 us to be as good as any that can be put into a 

 text-book. What makes it so unusually clear is 

 that the author proves in a separate article (§81) 

 that the condition for the compatibility of 

 f{x, y, z, p, q) = o, <f>{x, y, z, p, q) = 0, dz = pdx + qdy, 

 is [/, <^] =0, where the symbol on the left is that 

 introduced by Jacobi. Later on we have dis- 

 cussions of Cauchy's method (pp. 249-64) and 

 of Jacobi 's method (pp. 265-78). It should be 

 added that there is a very brief account (pp. 86- 

 98) of Lie's theory of transformation-groups. 



From time to time the author pauses to make 

 a general remark on this or that aspect of his 

 subject, and these obiter dicta deserve the most 

 careful attention. For instance : " Although this 

 reduction is not, in many cases, of any practical 

 utility, it nevertheless possesses great theoretical 

 interest, for it enables us to determine just how 

 difficult the problem is" (p. 214). Most text- 

 books on differential equations are very mislead- 

 ing, because they give the student the impression 

 that the subject is very much better understood 

 than it really is. The most simple-looking partial 

 differential equations may baffle the most eminent 

 mathematicians, and it would scarcely be too much 

 to say that there is no extensive theory of differ- 

 ential equations except for linear ordinary equations 

 the coefflcients of which are of certain specified 

 types. This assertion is not so paradoxical as it 

 looks ; all the fundamental functions of analysis 

 {not of arithmetic) can be defined by very simple 

 ordinary differential equations ; for instance, 

 exp(rv) is that solution of dy/dx = y which has the 

 value I when ,\; = o. All the properties of exp(.x) 

 can be deduced from this, and the whole of ana- 

 lytical trigonometry is then only a corollary. 



(2) Prof. Blichfeldt collaborated with Messrs. 

 G. A. Miller and L. E. Dickson in a work on 

 finite groups reviewed in these columns on 

 November 23, 1916 (vol. xcviii., p. 225). The 

 present work, dealing with collineation groups, so 

 far departs from abstract group-theory as to 

 choose a special imagery, or, if you will, a 

 drapery, for the sets of abstractions considered. 

 Every group may be imaged as a substitution 

 group ; not every group can be represented by a 

 collineation group. So Prof. Blichfeldt has re- 

 stricted his field of inquir\% and deliberately 

 tried not to use abstract group-theory any more 

 than he can help. For the purpose he has in 



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