NA TURE 



217 



THURSDAY, JANUARY 6, i{ 



CAYLEY'S PAPERS. 

 The Collected Mathematical Papers of Arthur Cay ley, 

 Sc.D., F.R.S. Vols, viii., ix. Pp. liv + 570, xvi + 622. 

 (Cambridge : at the University Press, 1895, 1896.) 



THESE two volumes form the first of those published 

 after Cayley's death in 1895. The first thirty- 

 eight sheets of Vol. viii. were revised by the author, who 

 added a note on one paper (No. 518) ; the duty of edit- 

 ing the rest of the papers was entrusted to Prof. Forsyth, 

 who has very faithfully carried out the plan and arrange- 

 ments which, in the absence of definite instructions, he 

 was able to infer from the previous volumes. 



Perhaps the reader's first impression after surveying 

 these 144 papers, mostly published in the years 1871-77, 

 is that they are very miscellaneous, and that compara- 

 tively few are of paramount importance. The fact is 

 that Cayley is, as it were, brought into unfavourable 

 comparison with himself ; short notes on special problems 

 of geometry and analysis, and solutions of Smith's Prize 

 papers cannot rank with the immortal " Memoirs on 

 Quantics," or some of the earlier geometrical papers, 

 such as that upon plane cubic curves. But it. is un- 

 reasonable to expect an artist to produce an uninter- 

 rupted succession of masterpieces ; and it is to be 

 remembered that Cayley seldom, if ever, wrote upon 

 any subject without developing some instructive point 

 or giving an example of his own characteristic elegance. 



In trying to give some account of the more important 

 of these memoirs it will be convenient to take the 

 geometry and the analysis separately. Not that the 

 boundary line is very easy to fix : Cayley was never a 

 geometrician in the sense in which the word may be 

 applied to Apollonius or Steiner. But some of the 

 papers have an interest mainly geometrical, although 

 the methods used are almost wholly algebraic ; and 

 with them we will begin. 



Perhaps the most important are those which deal with 

 transformation, correspondence, and the singularities of 

 algebraical curves and surfaces. With these difficult 

 theories Cayley dealt in a masterly way : he avoided, as 

 if by instinct, the many opportunities of mistake which 

 present themselves in a method which is largely enumer- 

 ative, and he had the gift of predicting general results 

 from the consideration of special cases. 



Coming next to what may be called the metrical 

 geometry of surfaces, which has developed so greatly 

 in recent years, we have papers on curves of curvature, 

 on geodesies on quadrics, and on orthogonal surfaces. 

 To this group may perhaps be added a paper on evolutes 

 and parallel curves, though this is rather meant to illus- 

 trate the non-Euclidian geometry. 



There are three monographs, on Steiner's surface, on 

 the centro-surface of an ellipsoid, and on the con- 

 figuration of the twenty-seven lines of a cubic surface, 

 which are in various ways highly characteristic. As 

 models of analytical skill they are admirable ; and as 

 helps to the understanding of the geometrical figures 

 with which they deal, they are of great service. But it is 

 curious to see how chary the author is in giving illus- 



NO. 147 1, VO... 57] 



trative diagrams. There are, indeed, two figures in the 

 paper on the surface of centres ; but why, we ask, did 

 Cayley not give a series of contour lines of the surface ? 

 or again, with still more reason, in the case of Steiner's 

 surface ? .Then the paper on the twenty-seven lines of a 

 cubic surface is so quaint in its topsy-turveydom as 

 almost to suggest Mr. W. S. Gilbert as joint author. 

 Here we have a projective configuration which may be 

 realised with the help of a bundle of sticks and without 

 any measurement whatever. What Cayley did was to 

 take a model by Dr. Wiener, measure approximately the 

 coordinates of a number of points upon it, thence find 

 the approximate equations of the lines, and finally adjust 

 the equations so as to satisfy the geometrical conditions ! 

 Of course there is reason in this seeming perversity : 

 by the projective method it is not easy to get a con- 

 venient arrangement of the sticks, whereas Cayley's 

 equations make it possible to construct a string model 

 on a cardboard frame without a tiresome series of pre- 

 liminary experiments. 



The poristic polygons of Poncelet appear to have had 

 for Cayley a perennial charm : we have here two papers 

 suggested by Poncelet's results ; one " On the porism of 

 the in-and-circumscribed polygon, &c.," which treats of the 

 original problem, and the other " On the problem of the 

 in-and-circumscribed triangle," which really deals with a 

 rather different and more general theory. Cayley, like 

 many others, does not seem to have been aware (at least 

 in 1871) that the complete algebraical solution of the 

 Poncelet problem was published in 1863 in a paper by 

 M. Moutard, which formed part of the appendix to 

 Poncelet's "Applications d' Analyse a la Geometrie." 

 Not only is this so, but, as Halphen pointed out, this 

 paper contains the first fully satisfactory treatment of 

 the multiplication of the argument in elliptic functions. 



Before passing on from the geometrical papers, atten- 

 tion should be called to the very interesting series of 

 notes on the mechanical description of curves. This is 

 a promising field of research, and the results could 

 hardly fail to be of interest, especially to those who like 

 to see the deductions of theory embodied in an actual 

 geometrical figure. There is an aesthetic satisfaction 

 in this contemplation : and, moreover, a really correct 

 figure often suggests geometrical truths that would 

 otherwise be overlooked. 



Of the analytical papers the one which has been most 

 appreciated in this country is, beyond question, the 

 short paper " On the theory of the singular solutions of 

 differential equations of the first order" {Messenger, 

 vol. ii. (1873) PP- 6-12). Here Cayley's power of giving 

 to analysis a geometrical interpretation appears to the 

 best advantage. If we have an algebraical relation 

 fix, y, p) =. o in which p enters to the degree s, then 

 this associates with any point ix, y) a series of s (real 

 or imaginary) directions corresponding to the different 

 values of / : in other words, the differential equation 

 really expresses that the plane of reference is covered 

 with 00^ tiny j-rayed stars. The primitive <^ {x, y, c) = o 

 gives a family of ooi curves each made up of ooi selected 

 rays. Now if we eliminate p from f{x, y, p) — o, 

 gy/ dp = o, we obtain a locus of points {x, y) at each of 

 which two rays coincide in direction ; where this happens 

 either two consecutive curves <^ {x, y, c) = o touch, or 



