NATURE 



441 



THURSDAY, SEPTEMBER 9, 1897. 



ABELIAN AND THETA FUNCTIONS. 

 AbeFs Theorem and the allied Theory^ including the 

 Theory of the Theta Functions. By H. F. Baker, 

 M.A. Pp. XX + 684. (Cambridge : at the University 

 Press, 1897.) 



CAYLEY'S often-quoted simile which compares the 

 province of mathematics with " a tract of beautiful 

 country seen at first in the distance, but which will bear 

 to be rambled through and studied in every detail of 

 hillside and valley, stream, rock, wood, and flower," 

 suggests a comparison of mathematical text-books with 

 those useful works which provide information and advice 

 for the tourist and the traveller. It may be said with 

 truth that there is every variety, from the cheap illus- 

 trated pamphlet, designed to catch the eye of the holiday 

 tripper in search of the picturesque, to the elaborate 

 maps, surveys, and gazetteers which are best appreciated 

 by the genuine explorer. 



It is to the latter class that Mr. Baker's treatise 

 belongs. It is in no sense a book for beginners ; in 

 order to appreciate it, the reader must be tolerably 

 familiar with modern function-theory, and not wholly 

 ignorant of the special subject of the work. Even then, 

 he will probably find the treatise most useful as a 

 "hand-book" ; that is to say, as a methodical guide to, 

 and commentary upon, the host of memoirs which are 

 ultimately connected with Abel's researches on algebraic 

 integrals. In this respect, the work is sure to be very 

 valuable ; for the author has evidently spared no pains 

 in making himself famiHar with everything of importance 

 that has been done in this field, and in attaining, so far 

 as possible, an impartial attitude towards the different 

 ways in which the subject has been presented. 



The title recalls the fact that not so very long ago 

 *' Abel's Theorem " was apt to be regarded by mathe- 

 matical students as being something like "Taylor's 

 Theorem" or "Ivory's Theorem": a stiff piece of 

 isolated analysis, for which one's " coach " was expected 

 to produce a concise and elegant proof, adapted for 

 writing out in an examination. Even now it may be 

 a surprise to some readers to find that a work pro- 

 fessing to deal mainly with Abel's Theorem can extend 

 to nearly seven hundred large and well-filled pages of 

 print. The truth is that while Abel's Theorem is in 

 itself a comparatively simple matter, and is perhaps best 

 regarded as a theorem in symmetric functions, it forms 

 a kind of centre from which several fundamental theories 

 may be said to radiate. In order to give precision to the 

 theorem itself, it is necessary to establish the real nature 

 and properties of algebraic functions ; Jacobi's problem 

 of inversion, or, in other words, the introduction of 

 Abelian functions properly so called, leads to the vast 

 and in some ways still mysterious theory of the general 

 theta-functions ; these, in their turn, have in late years 

 suggested an almost bewildering variety of transcendental 

 functions ; and, finally, we have the reaction of all these 

 discoveries upon analytical geometry. It is therefore 

 not astonishing that Mr. Baker's volume is so large ; it 

 NO. 1454, VOL. 56] 



would, in fact, have been much larger but for his studied 

 conciseness, which, indeed, here and there, may be 

 thought almost overdone. 



In a subject like this, which, perhaps more than any 

 other, has brought out the contrast between the in- 

 tuitional and analytical schools, it is always interesting 

 to observe an author's point of view. Mr. Baker has 

 adopted a kind of middle course which will commend 

 itself to those whose minds, like Cayley's, while essentially 

 analytical, love to clothe their discoveries in the language 

 of geometry. Thus the real start is made by assuming 

 the existence of Riemann's fundamental integrals of the 

 first, second, and third kind. Of course, after the work 

 of Neumann, Schwarz, and others, there is no logical in- 

 consistency in this ; and it is convenient for purposes of 

 exposition. But it is open to two objections : the first is, 

 that the proof of Riemann's existence-theorems is long 

 and difficult, when it is thorough ; the second, and more 

 essential, is that algebraic functions present themselves 

 as special collocations of integrals, and are not con- 

 sidered in the first instance on their own merits ; it is 

 hard to refrain from thinking that this is, to some extent, 

 putting the cart before the horse. 



But it ought in fairness to be said, that in subsequent 

 chapters (iii.-vii.), the analytical theory of Weierstrass, 

 Dedekind, Kronecker, and Hensel is explained in suffi- 

 cient detail to enable the reader to appreciate the other 

 point of view ; although, as a matter of fact, Weier- 

 strass's fundamental " Liickensatz " is deduced from the 

 existence-theorems. 



Points which deserve attention in these earlier chapters 

 are the careful explanation of "the places and infini- 

 tesimal on a Riemann surface " ; the discussion of the 

 Riemann-Roch theorem, which is unusually clear, and 

 well illustrated by examples ; and the satisfactory treat- 

 ment of adjoint polynomials. As to this last point, we 

 cannot but think that the advantage is all on the side of 

 the analytical school. They can explain why, and in 

 what sense, a singular point on a curve is to be reckoned 

 as S nodes and < cusps in the calculation of the deficiency; 

 can the intuitional mathematician say what 8 and k ought 

 to be at a higher singularity of a curve described by a 

 definite mechanical process, without finding the equation 

 of the curve ? 



Chapter viii. dismisses Abel's Theorem in the course 

 of twenty-seven pages. This brief treatment is made 

 possible by what has gone before ; still it is rather a 

 pity that more space has not been given to an inde- 

 pendent and purely algebraical treatment of at least the 

 differential form of the theorem. 



In chapter ix. we are introduced to the inversion 

 problem : and here it is pleasing to find an account of 

 Weierstrass's procedure, which is certainly the most 

 satisfactory in giving an intelligible form to the result, 

 although, of course, it is not a practicable method of 

 solving the problem. For this we must have recourse to 

 the theta-functions, and these are in fact introduced in 

 chapter x. It is with the theta-functions and their 

 properties that the rest of the book is principally con- 

 cerned ; chapters x., xi., xv.-xxi. are almost entirely 

 devoted to them, and give, in fact, the most elaborate syste- 

 matic treatment of the functions that has yet appeared 



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