704]
A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS.
509
82.
The product-theorem, and its results.
The product-theorem was
(a, /+
(u + u'). S-
ß'
7, 8'
u — u')
_ Vift i ( a + ®0 + P> 2 (ß + ß') + q ,n \ (a 2 ( a ~ °0 +P’ 2 (ß — ß') + 9.
s I S' .. ./ g _ g'
7 + 7
8+S
7“7
(2w'),
where only one argument is exhibited, viz. u + u', u— u!, 2u, 2u’ are written in place
of (u + u', v + v'), (u — u', v — v r ), (2u, 2v), (2u!, 2v') respectively. The expression on the
right-hand side is always a sum of four terms, corresponding to the values (0, 0),
(1, 0), (0, 1), and (1, 1) of (p, q). For the development of the results it was found
convenient to use the following auxiliary diagram.
Upper half of characteristic.
1
: t—1
r-H
r-H
t—1
H
r-H
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Hn
j r-lcc
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