X. ITU RE 



[November 6, 1902 



of each critical point ; that is to say, if near any critical 

 point a each integral can be put into the form 



y = (x- 



<Po+ <Pi log (■< - a) + . . . + (p,„ [ Iog(* - o)]» 



S 



where </>,„ </>,,... r/>„, are one-valued functions not infinite 

 at a. These equations are called by Prof. Forsyth 

 "equations of Fuchsian type." The equation of the 

 hypergeometric series is of this type, and is remarkable 

 as being the only one, of order higher than the first, 

 which is completely determined when the positions of 

 the critical points and the indices associated with them 

 are assigned. 



An equation of Fuchsian type may have one or more 

 algebraic integrals. If all the integrals are algebraic, 

 the group of the equation must be finite ; so here we 

 have a most unexpected concurrence of two apparently 

 disconnected theories. A very interesting problem is that 

 of determining linear equations the groups of which are 

 isomorphic with known finite groups ; another is that of 

 finding out whether a given equation has any algebraic 

 integrals. 



All the foregoing theory is discussed and illustrated by 

 Prof. Forsyth in a very attractive and lucid manner ; 

 thus chapter i. deals with the existence of a synectic 

 ntegral near an ordinary point and sets of independent 

 integrals ; chapter ii. with the expansions near a critical 

 point and with Hamburger's method of grouping them ; 

 chapter iii. with regular integrals ; chapter iv. with 

 equations of Fuchsian type ; and chapter v. with equa- 

 tions of the second and third orders possessing algebraic 

 integrals. Illustrations are supplied by the familiar 

 equations of mathematical physics, by the equation of 

 the elliptic quarter-period, and by that of the hyper- 

 geometric series. It is delightful to see how the dis- 

 cussion of these equations is illuminated by the general 

 theory. 



After a chapter on equations w th only some of their 

 integrals regular, we come to the consideration of 

 integrals with essential singularities. The most familiar 

 example of a function with an essential singularity at a finite 

 place is exp {x~ l ), which is the integral of x 2 / + y = o ; 

 and it is easy to see that if P is any polynomial in .r -1 , 

 the expression expP has an essential singularity and 

 satisfies a linear equation of the first order. 



Suppose now that we find that a given equation has an 

 integral with an essential singularity at the origin ; it 

 may be possible to express it in the form expP .xp\J/(x), 

 where p is constant and \Jr(.r) holomorphic. Such an 

 integral has been called "normal''; the discussion of 

 these integrals, and others obtained by putting x l l k for 

 x, is given in chapter vii., which contains important 

 results due to Thome, Hamburger, Poincare and others. 

 There is also a brief account of " double-loop integrals " 

 after Jordan and Pochhammer, and of Poincare's theory 

 of asymptotic integrals. 



In his paper on the motion of the moon, Hill was led 

 to the solution of a linear equation by a method in- 

 volving the use of infinite determinants. In chapter viii. 

 Prof. Forsyth discusses this method in some detail, after 

 giving a preliminary account of infinite determinants 

 and their properties. The subject of this chapter is not 

 very attractive in itself, but on account of its practical 

 NO. 1723, VOL. 6/] 



importance has naturally attracted a good deal of 

 attention. 



Chapter ix. deals with equations with uniform periodic 

 coefficients, and gives an account of this part of the 

 subject which ought to encourage young mathematicians 

 to read the original sources and experiment on their 

 own account. It is, of course, the equations with doubly 

 periodic coefficients that are most interesting. Thanks 

 principally to Hermite, Halphen and Picard, some ex- 

 tremely beautiful results have been already obtained in 

 this field, and there can be no doubt that others are 

 awaiting discovery. 



The last chapter of this volume, on equations with 

 algebraic coefficients, must have been very difficult to 

 write, and appeals mainly to the specialist. Its principal 

 topic is Poincare's celebrated theorem that the integrals 

 of any linear equation with algebraic coefficients can be 

 expressed by means of Fuchsian and Zetafuchsian 

 functions. As Prof. Forsyth justly remarks, we cannot 

 hope to make practical use of Poincare's theorem until 

 the analysis of automorphic functions has reached a 

 higher state of development. To this end the treatise 

 by Klein and Fricke, now in course of publication, will 

 doubtless contribute largely. 



In conclusion, it may be well to remark that this 

 volume is in great measure independent of its prede- 

 cessors, and that a great part of it will be quite 

 intelligible to junior mathematicians provided that they 

 know the elements of the theory of a complex variable. 

 To them, therefore, as well as to their seniors, this book 

 may be heartily commended. G. B. M. 



SCIENTIFIC PSYCHOLOGY. 



Gnuidziige dcr pliysiologischen Psychologic Von Wil- 

 helm Wundt. Funfte vollig umgearbeitete Auflage. 

 Erster Band. Pp. xv + 553. (Leipzig: W. Engel- 

 mann, 1902.) Price \os. net. 



THIS volume of 553 pages is the first of the three 

 volumes in which the fifth edition of Prof. Wundt's 

 great work is to appear. The rapid increase in size of 

 the work in each of the successive editions is thus main- 

 tained in the present one, and, as in the case of the 

 previous editions, has been necessitated by the rapidity 

 of the growth of the youngest of the natural sciences, 

 experimental or, as Prof. Wundt prefers to call it, 

 physiological psychology. And even the increase in 

 bulk of this book does not by any means fully express 

 the rate of growth of the science, a growth towards 

 which this country has contributed so lamentably little. 

 For the book is primarily a record of the work and the 

 views of the author and of his pupils in the great 

 Leipzig school. Nevertheless, Prof. Wundt has found it 

 necessary to rewrite almost the whole of the book, so that, 

 as he tells us, it must be regarded as almost a new one. 



The greater part of this first volume is concerned with 

 matters not strictly psychological, but rather with those 

 studies which form an essential part of the equipment of 

 the psychologist, namely, the fine and coarse anatomy, 

 the embryology and the physiology of nervous tissues, 

 both special and comparative. It is, perhaps, open to 

 question whether it is wise to attempt to treat so vast a 

 range of subjects in the scope of a single volume. For 



