240 Proceedings of Royal Society of Edinburgh . [sess. 
Some Identities connected with Alternants and with 
Elliptic Functions. By Professor W. H. Metzler. 
(Read June 2, 1902.) 
As is well known, the usual form of the Addition-Theorem for 
Elliptic Functions of several arguments expresses these functions as 
the quotient of two determinants. When two or more arguments 
become equal, both numerator and denominator of this quotient 
vanish, and in seeking to remove the common vanishing factor, 
Cayley, in his paper * “ ISTote sur l’addition des fonctions ellip- 
tiques,” in connection with the cases of three and four arguments, 
brought to light some identities connecting certain alternants. 
Cayley gave these identities without proof, saying, “ Je n’ai pas 
encore trouve la loi generale de ces equations.” 
Dr Thomas Muir, in his very interesting paper f of the same 
title as this one, gives several demonstrations of these identities, 
and ends by saying, “ The problem of finding for determinants of 
a higher order than the fourth, identities similar to Cayley’s, I do 
not at present enter upon ; like Cayley, ‘ Je n’ai pas encore trouve 
la loi generale de ces equations.’ I content myself with stating the 
problem in as simple a form as possible.” The problem for deter- 
minants of order five, as stated definitely by Muir, is to determine 
a, ft, y, 8, e, p, r ] , 0 , SO that 
(A) | a 0 b 1 c 2 D eE | . |A°B 2 C 4 D 6 ES| 
= | a° 6 1 c 2 D 2 eE 2 | • | A 0 B“ C 8 D y E 5 | 
± | <i° b 1 c 2 D 4 eE 4 | • | A 0 B e C p D’ E 8 | 
± | a° I? c 2 D« e& | ■ | A° B 1 C 2 I) 3 E 4 | 
where | b l c 2 D eE | stands for the determinant 
* Crelle's Jour., xli. pp. 57-65 ; or, Collected Math. Papers, i. pp. 540- 
549. 
t Trans. Roy, Soc. Edin., vol. xl. part i. (No. 9). 
