xli 



Abelian functions of two variables, and those of Weierstrass, developed 

 by Konigsberger* to give the " addition- theorem." Proceeding in 

 his ' Memoir on the Single and Doable Theta-functions 'f more by 

 (rcipel's method than by Roseuhain'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 Ku miner'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 conies, 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 circum- 

 scribed about the other, is due to the geometrician Poncelet. The 

 special case when the conies are two circles had been discussed 

 analytically by Jacobi,J using elliptic functions for the purpose. 

 Cayley undertook, first in 1853, the analytical discussion of the most 

 general case of two conies, also using elliptic functions ; and be 

 obtained 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 

 cases, that he did not leave the matter at this stage. In a memoir j| 

 * 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 com- 

 pletely 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 men- 

 tioned his investigations on attractions, specially those on the attrac- 

 tion of ellipsoids, to which he devotes five memoirs,^ discussing the 

 methods of Legendre, Jacobi, Gauss, Laplace, and Rodrigues ; and his 



* ' Crelle,' vol. 64 (1865), pp. 1742. 



t 'Phil. Trans.,' 1880, pp. 8971002. 



J ' Ges. Werke,' t. 1, pp. 277 293 ; this paper waa published first in ' Crelle,' 

 t. 3 (1828), pp. 376389. 



In a set of fire papers, 'C. M. P.,' vol. 2, Nos. 113, 115, 116, 128; ibid. 

 vol. 4, No. 267. 



|| <C. M. P.,' vol. 8, No. 514; ' Phil. Trans.' (1871), pp. 369412. 



1 ' C. M. P.,' vol. 1, Nos. 75, 89; vol. 2, Nos. 164, 173, 193. 



TOL. LVIII. e 



