lyo 



NATURE 



[June 22, 189: 



Weierstrass, to whom it is due, has never been printed ; 

 and the only published proof, besides the one which Dr. 

 Forsyth gives, appears in a paper by M. Phragmen in 

 vol. vii. of the "Acta Mathematica," and is on entirely 

 different lines. Whether either proof is entirely satis- 

 factory is a point on which differences of opinion may 

 conceivably occur, though of course there is no doubt as 

 to the truth of the theorem itself. 



Chap, xiv., which is headed " Connectivity of Surfaces," 

 is purely geometrical, and strictly has nothing to do with 

 the theory of functions. It was however necessary for i 

 the author to introduce such a digression if the following : 

 chapters dealing with Riemann's theory were to be under- 

 stood, since there is no treatise to which reference 

 could be made for the various theorems and results that 

 have to be used. The chief properties of a Riemann's 

 surface, regarded as arising from an algebraical equation 

 between the variables, are discussed in Chap. xv. Though 

 there is no difficulty in conceiving the geometrical nature 

 of a Riemann's surface from a description, the relation 

 between the surface and the set of functions (algebraische 

 Gebilde) whose study it is intended to simplify is not so 

 readily grasped at first by the student ; and it would not 

 perhaps have been amiss to have dealt with this relation 

 in one or two simple cases, at some length, as an intro- 

 duction to this part of the subject. In Ch.ap. xvi., the 

 surface still being regarded as defined by a given 

 equation, the properties of uniform functions on the 

 surface, and of their integrals, is investigated. 



From this point to the end of the book we have to 

 do, more or less directly, with the fundamentally new 

 conception of Riemann which has been so wonderfully 

 developed during the last ten or fifteen years. The 

 Riemann's surface, as defined by a given equation, affords 

 a most convenient means of study of a system of con- 

 nectedfunctions. Suppose,however,the surface tobe given 

 quite independently of any equation. The possibility at 

 once suggests itself that the surface may serve as the 

 definition of a set of connected functions. Riemann's 

 own demonstration that this is the case has since 

 been shown to be faulty, but the conception is an 

 invaluable one, and it has been placed on a secure founda- 

 tion by Schwartz (and others), by means of the so-called 

 existence theorem. Chap. xvii. is entirely occupied with 

 the proof of this theorem, and in Chap, xviii. follow the 

 investigations with respect to the form and nature of the 

 integrals and uniform functions, so shown to exist, on a 

 Riemann's surface given arbitrarily. 



Chaps, xix. and xx. deal at length with the theory of 

 conformal representation. This forms one of the most 

 obviously interesting parts of the subject, and is also one 

 of those which lend themselves most readily to the purposes 

 of application ; and it is to be noted that, although owing 

 to necessities of arrangement these chapters occur near 

 the end of the book, the author suggests that, on a first 

 reading, Chap. xix. should be taken at an early stage. 



The last chapter in the book gives an introduction to 

 the theory of automorphic functions, the previous one 

 being taken up by a necessary digression on groups of 

 linear substitutions. Dr. Forsyth follows M. Poincard in 

 actually obtaining analytical expressions for the functions 

 in the form of the ratio of infinite series, analogous to the 

 expressions for elliptic functions as ratios of the theta- 

 NO. 1234. VOL. 48] 



functions. These analytical expressions, though of great 

 interest, are too complicated in form to be readily used for 

 deducing the properties of the functions they represent, 

 so that their properties must be inferred from their 

 quasi-geometrical definition by means of a " fundamental 

 region"; and this is essentially the method of dealing 

 with them used by Prof. Klein. 



In thus shortly stating the contents, or rather the head- 

 ings, of the successive chapters some risk is run of repre- 

 senting the book as a mere compilation. Nothing could 

 possibly be further from the truth. From the nature of 

 the case it is inevitable that the greater portion of the 

 book should be taken up with detailing the results of other 

 writers, but Dr. Forsyth has done this in a most inde- 

 pendent way. The book is instinct all through with an 

 original spirit ; in numerous instances, where clearness 

 or conciseness were to be gained, the author has 

 modified or completely altered the usually-given proofs, 

 while, as has been already stated, the various parts of the 

 subject have been brought together, and the many 

 different ways of dealing with them have been used, in 

 such a way that the theory is presented to the reader as 

 a connected and harmonious whole. Dr. Forsyth is to 

 be warmly congratulated on having brought to so success- 

 ful a conclusion what must have been an extremely 

 arduous task. If it is not ungracious to " ask for more ' so 

 soon, we may express the hope that he will now go on to 

 deal, as completely and successfully, with functions 

 defined by differential equations. 



The book itself is beautifully printed and the figures, 

 many of which must have required careful drawing, are 

 well reproduced. The table of contents is sufficiently 

 complete to'.form a sort oiprc'cis of the whole ; and lastly, 

 we have to be grateful for three separate indices. The 

 first of these, an index to all the technical terms used in 

 the book, whether English or foreign, is a most useful 

 addition ; especially for those who wish to use the book 

 without reading right through it. W. Burnside. 



TINCTORIAL ART AND SCIENCE. 

 A Manual of Dyeing: for the use of Practical Dyers y 

 Manufacturers, Students, and all interested in the 

 Art of Dyeing. By Edmund Knecht, Ph.D., Chris- 

 topher Rawson, F.I.C., and Richard Loeweathal, 

 Ph.D. (London : Charles Griffin and Co., 1893.) 



THE present work consists of three volumes, two of 

 letterpress, interspersed with illustrations of plant, 

 which run to over 900 pages, and a third volume con- 

 taining specimens of dyed fabrics. It is a substantial 

 contribution to an important branch of technology, and 

 the authors have succeeded fairly well in meeting the 

 requirements of the various classes of readers for whose 

 use the work has been written. The first general im- 

 pression produced on looking through the volumes is one 

 of satisfaction that the subject is handled in a more 

 scientific way than has hitherto been the case in such 

 works. The only feeling of disappointment to which 

 the consideration of the book gives rise is in no way 

 attributable to the authors, but is due to the circumstance 

 that so little is known about the scientific relationship- 

 between a colouring-matter and the fabric which is dyed 

 thereby. All that is known about the theory of dyeingf 

 is ably stated in the introductory chapter, and one of the: 



