854 
MR. A. E. FORSYTH OH THE THETA-FUNCTIONS, 
63. Each of these products is affected with a positive or negative sign, determined 
by the following rule (Buie I.); and it has this sign modified by being multiplied, 
according to another rule (Buie II.), by a definite power of negative unity. Taking 
the first term, viz.:— 
11® 
Ap At,, . . ., AA-y. 
lU, N 3 , . . . , N rr 1 ’ 
the characteristics for the remaining 4 r —1 similar products may he written down as 
follows : for the upper rows take all possible combinations of 
r A 15 r a 2 , Ag, r A Sf 
[Ax + l, [A^ + l, [Ag + l,. .., [Ai+1,. . 
JA,, 
,IA+i, 
by selecting one from each bracketed pair; and similarly, for the lower rows, from 
JNp JN 3 , [N 3 , JN„ JN „ 
t^ + l, |N 2 +1, 1N S + 1, . .. ,LN,+1,..., |Nr+l. 
Defining that number in the lower row of the characteristic as “ corresponding ” 
with another in the upper row (and vice versa) when the two have the same suffix, 
the following are the rules above referred to :— 
Buie I. If, in the typical characteristic of any product, there be an odd number of 
pairs of corresponding numbers such that each member of a pair differs by unity from 
the member holding the same position in the first term, viz.: in 
IBE> 
/A v A 2 , 
\x 15 n 2 , 
X 1? X, 
then to that product is prefixed a negative sign; but if there be an even number of 
such pairs, a positive sign must be prefixed. 
(As in the algebraical expression of the theorem the numbers will be of the form 
JST s +l, Nj+1 and not of the form N s — l=N s -f 1 — 2, N 4 — l = N t +l — 2, which might 
by formula (220) cause a difference of sign, the above rule is perfectly determinate). 
Buie II. To find the index of (—1)* in order to prefix the proper power of (—1) to 
a product, there must be taken the sum of the numbers in the upper rows of the 
characteristics of the four functions in the first term, viz.: in 
n<£ 
r/Ap a 2 , . 
LWp x 2 , . 
A, 
N, 
X 1? x 2 , 
corresponding to those in the lower row which hold the same position as the numbers, 
differing from them by unity, in the lower row of the typical characteristic of the 
product. 
