50 



NA TURE 



[May 1 9, 



glad to be out of this plague-stricken place. The greater 

 part of them died, and they took the plague to England, 

 and they have not got rid of it since." 



The book is well illustrated by reproductions of old 

 prints and pictures and drawings of the places as they 

 exist to-day. It is one of the most entrancing studies 

 we have met with, and can be read over and over again. 

 We heartily congratulate Mr. Paget on his work. 



CAYLEY'S MATHEMATICAL PAPERS. 

 The Collected Mathematical Papers of Arthur Cay ley., 

 Sc.D., F.R.S. Vols. X., xi. Pp. xiv + 6i6 ; xvi + 644. 

 (Cambridge : at the University Press, 1896.) 



THIS instalment of the papers illustrates in a remark- 

 able way Cayley's power of commenting upon and 

 developing the work of his predecessors. The various 

 memoirs on single and double theta-functions are, of 

 course, based upon the results of Rosenhain, Gopel, and 

 Kummer ; and it is instructive to see how Cayley's instinct 

 for symmetry and logical consistency has enabled him to 

 present the theory in a compact and intelligible form. In 

 the case of the single theta-functions, defined by their 

 expansions in series, we have equations such as 



e'oAo(« + z')^oo(«--^) = «'oo(")»'oo(^^) + ^-ii(«)9'n(^') ■ • ■ (i-) 

 and from these it appears that any three of the squared 

 functions 6'^jrh{u) are connected by a linear relation. 

 Hence we may take the squared functions to be propor- 

 tional to A{a-x)^ B(^-x), C{c-x), Y>{d-x) with x a 

 variable, and the other quantities constant. Finally it is 

 shown that x and u are connected by a differential equa- 

 tion of the form 



du = -7:^ 



W{a-x)(i-'X){i--x){d-x) 



Proceeding next to the double theta-functions, Cayley 

 gives a set of 256 equations analogous to (i.) ; from these 

 are derived quadric relations between the 16 functions 

 which give, in all, 72 asyzygetic relations ; it is assumed, 

 and is fairly evident, that these are all the independent 

 relations. The existence of the Kummer hexads and 

 Gopel tetrads gives a special character to these relations. 

 The next step is to find algebraic functions of two vari- 

 ables X, y and a proper number of constants which, on 

 being substituted for the 16 theta-functions, satisfy the 

 quadric relations identically. This Cayley succeeded in 

 doing, apparently by a series of happy guesses ; and this 

 is his main contribution to the theory. He also shows 

 that the two sets of variables u, v and x, y are connected 

 by differential relations of the form 



where CT, p, a, t are constants, X = (a - x) {b - x) . . . 

 (/-jt), a sextic \w x, and Y is the same function of _y 

 that X is of X. 



In order to complete the theory, from this point of 

 view, it is necessary to find the connection between the 

 constants which occur in the theta-functions as originally 

 defined and those which are contained in the correspond- 

 ing algebraical expressions. This can, in fact, be done 

 NO. 1490, VOL. 58] 



adu + Tdv 



- Vx/X 



for the single theta-functions (vol. x. p. 482) ; Cayley 

 began, but did not finish the corresponding investigation^ 

 for the double theta-functions {ibid., pp. 563-564). 



It would probably be well worth while to work out the 

 relations of Cayley's theory to recent researches on 

 hyperelliptic sigma-functions by Klein, Burckhardt and' 

 others. The best general view of Cayley's results is to 

 be found in the " Memoir on the Single and Double 

 Theta-Functions " (No. 704). 



Suggested by the theta-function theory, there are 

 several important geometrical papers, as, for example, on 

 the i6-nodal quartic surface, and on the bitangents of a 

 plane quartic. 



The memoir " On the Schwarzian Derivative and the 

 Polyhedral Functions " is chiefly valuable for its detailed 

 analytical work, which is a great help to the proper ap- 

 preciation of the papers of Kummer and Schwarz, 

 especially the latter. In this connection it is proper to 

 mention Cayley's own papers on the correspondence of 

 homographics and rotations and on finite groups of linear 

 substitutions (Nos. 660, 752). 



Of the other papers on group-theory the most important 

 is No. 690 ; this contains the "colour-diagram," and the 

 maxim, adopted by Dyck as the motto of his " Gruppen- 

 theoretische Studien " : " A group is defined by means of 

 the laws of combinations of its symbols." This ultimate 

 symbolical form of a group is, so to speak, its transcend- 

 ental essence, which may become incarnate in an endless 

 variety of shapes, such as sets of permutations, geo- 

 metrical configurations, motions in space, and so on. 



In the region of pure algebra we may notice the tenth 

 memoir on quantics, which gives a very complete 

 account of the binary quintic ; tables for the binary sextic 

 and ternary cubic ; and a paper on the Jacobian sextic 

 equation. 



Vol. xi. contains a reprint of the articles contributed by 

 Cayley to the " Encyclopcedia Britannica." These, per- 

 haps, will convey to the general reader some sense of his 

 characteristic qualities as a writer ; clearness, order, 

 philosophical breadth and independence of view, com- 

 bined with a studied restraint of manner which some- 

 times inclines to coldness. This reserve arose, probably, 

 from an excess of sensitiveness, which made him follow 

 an ideal of classic severity and shrink from any open ex- 

 pression of emotion. That he fully appreciated the 

 aesthetic side of mathematics is clear from the well-known 

 passage in his presidential address to the British 

 Association, where he describes the extent and variety of 

 modern mathematics by a metaphor of great beauty and 

 appropriateness. But this is a rare, if not solitary excep- 

 tion to his usual custom ; to gain a true idea of his 

 personal charm we must appeal, not to his published 

 work, but to the testimony of the friends who knew him 

 well. For them the portrait prefixed to vol. xi., which 

 shows Cayley as he was in 1885, will form a touching 

 memorial. 



Of the numerous minor papers, and of the problems 

 and solutions contributed to the Educational Times, it is 

 needless to say anything here. Diamond-dust from the 

 lapidary's workshop, they will doubtless help to polish 

 gems not yet extracted from the mine. G. B. M. 



