858 
MB. A. B. EOBSYTH ON THE THETA-FUNCTIONS. 
66. From an inspection of these equations, it is seen that the lower row in the 
characteristic of each term is the same throughout the same side of the same equation ; 
and this holds throughout the system of 2 r equations of which the above are examples. 
To write down the equation in which k of the numbers in the lower row on the right- 
hand side differ by unity from N 1? N 3 , . . . , N r , (and which is therefore an equation 
comprising —— : —- cases), take that group of 2 r terms in the general product theorem 
having this lower row for the common lower row of the characteristic and multiply the 
group by 
( _ 1 ^S2A< 
t 
where SA t has already been defined and S implies that summation is to be taken for 
those k values of t which have their numbers in the lower row of the characteristic of 
the form N ( +1 ; this is the right-hand side of the equation. To obtain the left-hand 
t 
side the coefficient ( —l)" S2A( is dropped, as well as all the units in the numbers 
N t +1, • . . ; and X, v, x are substituted for A, N, X respectively. Thus to a term of 
the form 
|i{2Ai+2A ffl +-2A n +SA (/ )rr^ J i &-V A 3 , A 3 + 1. , . . , A K +1 , . . . , A? +1, . . . , 
,...,^ + 1,..., 
X//. 11> * ■ > > Xu ! • ' • j A-p “t" 1J • • . } A q , . . A 
X,„ +1,. . ., X„ 4-1. . . . , Np , . . . , -f 1,. . . , N 
x 3 , . .. , X,} 
on the right-hand side, there will on the left-hand side be a term of the form 
l\ v l’ v v v z , . - . >V K , . . . , Vi , 
, 7v, w 4-1, . 
5 Vm i ■ 
. . . , X p +1, . . . , 
V)h . • * > Vp }••• > Vqr> 
X 
n 
67. By increasing each variable x (and therefore also X, from its definition) by the 
quarter quasi-period in any set of conjugate quasi-periods, and taking all combinations 
of these (amounting in number to 2 r ), each equation of the above system of 2 r equations 
gives rise to 2 r further equations. The reason of this is that each function is periodic 
with the exception of a factor 
iirx* 
and therefore a product on the left-hand side is periodic with the exception of a factor 
