784 
MR. A. R. FORSYTH ON THE THETA-EUNCTIONS, 
\( x )> 0 V ,M being single theta-functions. From it are also obtained the expressions 
for the four periods, as well as a proof of the product theorem of Section I.; and the 
function <3> is shown to satisfy two differential equations of the form 
d*<f> 
~d^ 
— 2xl K 
jz _A 
K!^ +2KK h x = 0 
( K > k', E having the ordinary meaning in reference to 6 jh K (x)), and an equation of the 
form 
dg> 2KA d 3 <S> 
dr 7 r 3 dxdy 
Section III. forms the expression of the addition theorem. Although no addition 
theorem proper exists for theta-functions (that is to say, 0(cc+f, y-\-y) cannot be 
written down in terms of functions of x, y and of f, rj ), an expression is obtainable for 
$(»+£ y+y) • £ y—y) 
<3>, being either the same or different functions. Since any one function of the sum 
may be combined with any function of the difference of the variables, 256 equations 
are necessary; and these are written down in 16 sets of 16 each. 
In Section IY. many of the properties already proved for the double theta- 
functions are generalized for the “ r ” tuple functions. Among these are :— 
(i.) The periodicity as in Section I.; 
(ii.) The product theorem, which gives the product of four functions as the sum 
of i r products of four functions ; from it several general relations are deduced ; 
(iii.) The analogue of the theorem in Section II., viz.:— 
3> 
■ \\ i 
Pi, ■ ■ ‘> %r\ 
1 W ^ * • 
■ M J 
o s—r t~r 
--2 2 K,K( log ^ < 
= e 7r «=i 
O^x,) 
(iv.) The r differential equations of the form 
dx r 2X \ Kr I< Jdx,T ' r d Kr ~ 
and the \r(r—l) of the form 
d<S> , 2K s Kt d*® 
P Wl\t + fcfc* 
= 0 
all satisfied by <P. 
