442 



NATURE 



[September 9, 1897 



in English. The conciseness of the notation and the 

 difficulty of the subject make these chapters very hard 

 reading. Every one acquainted with the subject is aware 

 that the multiple theta-functions sprang from a brilliant 

 generalisation of Jacobi's functions e(«), H(«)) and 

 that the crux of the whole matter is to establish, 

 in a natural way, their connection with a set of alge- 

 braical functions. It is impossible to feel that this con- 

 nection has been made in the really proper and natural 

 way : the theta-functions drop out of the clouds, so to 

 speak, and are joined up to the inversion problem in a 

 more or less artificial and unsatisfactory manner. The 

 most promising glimpse of a better method seems to be 

 afforded by the " Prime Form " introduced by Schottky 

 and Klein ; and it is very satisfactory that due prom- 

 inence is given to this function in chapter xiv. (on fac- 

 torial functions) and elsewhere. It is very important to 

 notice that this function admits of an independent defini- 

 tion, and that the three kinds of elementary Riemann 

 functions may be directly and simply derived from it 

 (see Arts. 233-5). Moreover there is a relation con- 

 necting any theta-fuilction with a corresponding prime 

 form (Arts. 237, 272-4). Thus the prime form has a 

 strong claim to be considered fundamental ; and it is not 

 improbable that, by starting with it, the same kind of 

 simplification may be attained as that which has been 

 achieved in the theory of elliptic functions by the use of 

 Weierstrass's sigma-function. 



It would be unprofitable to attempt to give an abstract 

 of the chapters on the theta-functions which Mr. Baker's 

 book contains ; enough to say that (besides the elementary 

 properties) the relations among theta-products, the theory 

 of transformation, and the theory of characteristics are 

 all fully treated ; account being taken not only of memoirs 

 already classical, such as those of Rosenhain, Gopel, 

 Hermite, and Weierstrass, but also of quite recent re- 

 searches, such as those of Prym, Frobenius, Poincare, 

 and others. A certam amount of special attention is 

 given to the hyper-elliptic case ; this is justified by its 

 comparative simplicity, and the large amount of literature 

 connected with it. 



Some parts of the book, especially the last two 

 chapters (on complex multiplication of theta-functions, 

 the theory of correspondences, and degenerate Abelian 

 integrals), deal with problems of which the complete 

 solution is still the object of research. This is a very 

 welcome feature ; for the unavoidable incompleteness of 

 these parts is more than compensated by their stimu- 

 lating quality. It is a pity that authors of mathematical 

 treatises too often neglect the opportunity of carrying on 

 a discussion to actual contact with current research, and 

 pointing out the possible or probable direction of future 

 development. 



One characteristic of Mr. Baker's treatise seems to call 

 for remark. On p. 93 the author says : " We desire to 

 specify all the possibilities " ; this sentence might have 

 been adopted as a motto for the whole work. There is 

 such a wealth of conscientious detail that we can imagine 

 some readers failing to grasp the general argument, and 

 becoming disheartened by the array of complicated 

 formulae with their plentiful adornment of suffixes. Of 

 course, in order to treat the subject generally, a certain 

 amount of complexity is unavoidable ; there is, however, 

 NO. 1454, VOL. 56] 



an alternative which can often be adopted, and is worth 

 considering, namely, instead of introducing n symbols, 

 say ;»:„ x.> . . . x,„ to use a limited number, say .r, .r, x.^ or 

 X y z^ and to give the demonstration in such a form that 

 its generality can be inferred. It must be admitted that 

 this course is not always practicable ; the fact is that, in 

 order to read modern analysis with comfort, a certain 

 facility in handling sums and products in a condensed 

 notation is almost indispensable, and should be acquired, 

 if possible, at an early age, as in the analogous case of 

 definite integrals. 



The book contains a considerable number of illustrative 

 examples, many of which are worked out in detail. These 

 cannot but be of great help to the reader, by showing 

 how the general theory is brought to bear upon particular 

 cases. This is especially true in the actual construction 

 of the Riemann integrals for a given plane curve. 



Printers' errors appear to be very rare ; on p. 138, 

 line 14 from the bottom, "not greater than Q -^" 

 should be "not less than Q - p" ; and in the early part 

 of Art. 176 there are several misprints, which, how- 

 ever, the reader can easily correct for himself. 



There can be no doubt that this work will be highly 

 appreciated by all who make a special study of Abelian 

 functions ; and we trust that, in the approbation of the 

 limited circle to which he appeals, Mr. Baker will find 

 a sufficient reward for the immense amount of labour 

 which his task must have entailed. There is still room 

 for a strictly introductory work, bringing out the salient 

 features of the theory, and perhaps not disdaining 

 "heuristic" methods of investigation. In spite of its 

 limited scope and occasional diffuseness, there is a 

 charm about C. Neumann's book which we miss in the 

 more analytical treatises ; something of this kind in 

 English would probably do much to draw attention to 

 this very fascinating field of research, and induce a 

 select few to follow up the somewhat abstruse analysis 

 which a more detailed study of the subject involves. 

 Some reference to the historical evolution of the theory 

 would not be out of place ; indeed, we rather regret that 

 the plan of Mr. Baker's treatise has tended to obscure 

 this side of the matter. Cauchy is only mentioned once, 

 and Puiseux not at all ; yet the work of these two mathe- 

 maticians was fundamental, and will always form a part 

 of any systematic discussion of function-theory. 



G. B. M. 



THE CULTURE OF FRUIT. 

 The Principles of Fruii- Growing. By L. H. Bailey. 

 Pp. xi + 508. (New York : The Macmillan Company. 

 London : Macmillan and Co., Ltd., 1897.) 



FRUIT-GROWING in this country is one of the 

 remedies proposed to counterbalance the effects 

 of low prices for agricultural products. But fruit-growing 

 is an art which cannot be learned without experience. 

 It is no easy matter for a farmer to change his habits 

 and his practices, even if the local conditions are favour- 

 able to the production of fruit ; and the orchard of the 

 farm is very generally the most neglected part of the 

 whole establishment. Nevertheless, it is obvious that 

 a great extension of fruit culture has taken place during 



