242 
[753 
753. 
ON A THEOREM RELATING TO THE MULTIPLE 
THETA-FUNCTIONS. 
[From the Mathematische Annalen, t. xvii. (1880), pp. 115—122.] 
I propose—partly for the sake of the theorem itself, partly for that ot the 
notation which will be employed—to demonstrate the general theorem (3'), p. 4, of 
Dr Schottky’s Abriss einer Theorie der Abel’sehen Functionen von drei Variabein, 
(Leipzig, 1880), which theorem is there presented in the form : 
rii 1 *!!’ 
v) 
0(m 1 + 2® 1 / ,...; ¡i, v) = e 
- 27ri2/u. a »'' a 
0 (lij 
fM + fl', V + v), (S') 
but which I write in the slightly different form 
exp. [— H (u ; //, v')]. 0 (u + 2ot / ; p, v) = exp. [— 27ri/xv'] . © (u ; p + p, v -I- v). 
I remark that the theorem is given in the preliminary paragraphs the contents 
of which are, as mentioned by the Author, derived from Herr Weierstrass: and 
that the form of the theta-function is a very general one, depending on the general 
quadric function 
G(iii, iipi 
of 2p variables, p being the number of the arguments iq, ...,u p (in fact, the periods 
are not reduced to the normal form, but are arbitrary) ; and the characters v 1} ..., v p ; 
fii, ...,/jl p , instead of having each of them the value 0 or 1, have each of them any 
integer or fractional value whatever. The meaning of the theorem (u denoting a set 
or row of p letters iq, ..., u p , and so in other cases), is that the function 
0 (u ; u + p!, v + v)