753] 
ON A THEOREM RELATING TO THE MULTIPLE THETA-FUNCTIONS. 
247 
and the value therefore is 
= 4a (bt it + b/ 2 ) + 2h (2 j ar'v — itv') — b (2v v + v' 2 ) — 2iri (v + v) p!, 
On the right-hand side, putting the term in h under the form 
— 2 h (if + b/) v + 2Ab/ (2i< + v), = — 2 (/i) v (u + b/) + 2/ib/ (2v + v’), 
and the last term under the form 
— Trip (2v + v) — Tri/Jb'y', 
the equation becomes 
H (u; p!, v) = (4cm' — 2 (h) v) (u + b/) — iriplv 
+ (2/fBT — bv — 7rifl') (2v + j/), 
where the second line vanishes in virtue of the foregoing equation 
2liz?' - bv — 7rip! = 0 ; 
the equation thus is 
H(u; p!, v) = — 2 (h) v) (u + b/) — rrip!v, 
which equation, regarding therein b/ as a linear function of p! and v, shows that 
H (it; ¡x, v) is a function linear as regards u (and containing this only through it + «/), 
but quadric as regards p, v. 
Introducing the new row-letter we ma y write 
H (u ; ¡x, v) = 2£' (u + bt') — irifxv, 
viz. the expression on the right-hand side is here assumed as the value of the 
function 
H (it; fx', v), = G (it + 2b/, v)—G (it, v + v')- 2iri (v + v) p!; 
and the theorem then is 
exp. [— H (u; ¡X , v')\. © (u + 2bt 7 ; p, v) = exp. [— 27ripv]. © (u; p + p, v + v), 
where, by what precedes, 
2Ab/ — bv' — Tiifx' = 0, 
2aw' — (h) v — = 0, 
2p equations for determining the 2p functions bt 7 , £ 7 as linear functions oi fx, v : 
which equations depend on the p (2p +1) constants a, b, h. 
Suppose that the resulting values of b/ and £ 7 are 
b/ = cop + wv, 
£ = «//*' + 
where a>, &>', y, y are square-letters; then, regarding a, b, h as arbitrary, the 4p°- 
new constants w, w, y, y' cannot be all of them arbitrary, but must be connected 
by 4p 2 - p(2p + 1), =p(2p- 1) equations.