584 
[881 
881. 
ON HEEMITE’S iJ-PEODUCT THEOEEM. 
[From the Messenger of Mathematics, vol. xviii. (1889), pp. 104—107.] 
I give this name to a theorem relating to the product of an even number of 
Eta-functions, established by M. Hermite in his “Note sur le calcul differential et le 
calcul integral,” forming an appendix to the sixth edition of Lacroix’s Differential and 
Integral Calculus, and separately printed, 8vo. Paris, 1862. It is the theorem stated 
p. 65, in the form 
Hz 
<f)(x) = F (z 2 ) + zF-l (z 2 ), 
where 
^ /„a _ A3(tc - O H(x - a 2 ) ... H(x — a m ) 
<P W ©2« 
where a 1 + a 2 + ... + a 2n = 0, and £ = snx, cnx or dnx at pleasure; F(z 2 ), Ffz 2 ) denote 
rational and integral functions of z 2 of the degrees n and n - 2 respectively; A is a 
constant, which we may if we please so determine that in F(z 2 ) the coefficient of the 
highest power z 2n shall be = 1. 
If, for shortness, we write s, c, d for sn x, cn x, dn x respectively; and to fix the 
ideas, assume z — sn«, =s, then the theorem is 
AH(x — a l )H(x — a 2 )... H(x — a m ) 
0 m (x) 
= F (s 2 ) + scdF 1 {s 2 ); 
viz. the theorem is that the product of the 2n i7-functions (a x + a 2 + ... + a 2n = 0 as 
above), divided by © 2m («), is a function of the elliptic functions sn, cn, dn, of the 
form in question. 
Hermite uses the theorem for the demonstration of Abel’s theorem, as applied to 
the elliptic functions; or as I would rather express it, he uses the theorem for the 
determination of the sn, cn, and dn of + a 2 + ... + a 2M ._ 1 .