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xlii 
BIOGRAPHICAL NOTICE OF ARTHUR CAYLEY. 
years he wrote a number of papers dealing with the theta-functions as on an inde 
pendent basis and not as a detail in elliptic functions. Though the investigations are 
concerned with p-tuple functions, yet, partly for simplicity, and partly in order to secure 
the greater detailed development of the theory, the papers deal chiefly with the cases 
p = l, p = 2. 
Previous to Cayley’s investigations, the most valuable algebraical results in this sub 
ject were those of Rosenhain* and Gopel*}* which had connected the double theta-functions 
with the theory of the Abelian functions of two variables, and those of Weierstrass, deve 
loped by Königsberger| to give the “addition-theorem.” Proceeding in his “Memoir on 
the single and double theta-functions ”§ more by Gopel’s method than by Rosenhain’s, 
Cayley resumes the whole theory. He pays special attention to the relations among 
the squares of the functions and to the derivation of the biquadratic relation among 
four of the functions, which is the same as the equation of Kummer’s sixteen-nodal 
quartic surface. To this relation and to the geometry of this associated surface he 
frequently recurred, both specifically in isolated papers and generally in researches upon 
quartic surfaces. 
As connected, in part, with elliptic functions, his investigations on the porism of 
the in- and circumscribed polygon should be mentioned. The porismatic property of two 
conics, viz. that they may be related to each other so that one polygon (and, if one 
polygon, then an infinite number of polygons) can be inscribed in one and circumscribed 
about the other, is due to the geometrician Poncelet. The special case when the conics 
are two circles had been discussed analytically by Jacobi ||, using elliptic functions for 
the purpose. Cayley undertook, first in 1853, the analytical discussion of the most 
general case of two conics, also using elliptic functions; and he obtained IT the relations, 
necessary for the porism, for the several polygons as far as the enneagon. And it may 
be remarked, as a characteristic instance of Cayley’s habit of proceeding to general 
eases, that he did not leave the matter at this stage. In a memoir** “On the 
problem of the in- and circumscribed triangle” he raises the question as to the number 
■of polygons which are such that their angular points lie on a given curve or given 
curves of any order and their sides touch another given curve or given curves of any 
class. Using the theory of correspondence, he solves the question completely in the 
case of a triangle—taking account of the fifty-two cases that arise through the 
possibility of two curves, or more than two curves, being one and the same curve. 
From time to time Cayley turned his attention to questions in theoretical dynamics, 
choosing them as subjects of his lectures during his earlier years as professor. Among 
them may be mentioned his investigations on attractions, specially those on the attraction 
of ellipsoids, to which he devotes five memoirsj*j*, discussing the methods of Legendre, * * * § ** 
* Mem. des Sav. Etr. vol. xi. (1851), pp. 361—468; the paper is dated 1846. 
+ Crelle, vol. xxxv. (1847), pp. 277—312. 
X Crelle, vol. lxiv. (1865), pp. 17—42. 
§ Phil. Trans. 1880, pp. 897—1002. 
|| Ges. Werke, vol. i. pp. 277—293 ; this paper was published first in Crelle, vol. in. (1828), pp. 376—389. 
1 In a set of five papers, C. M. P. vol. ii. Nos. 113, 115, 116, 128 ; ibid., vol. iv. No. 267. 
** C. M. P. vol. viii. No. 514; Phil. Trans. (1871), pp. 369—412. 
ft C. M. P. vol. i. Nos. 75, 89; vol. n. Nos. 164, 173, 193.