0, 
0, 
0, 
0. 
704] 
A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS. 
547 
118. The foregoing equations may be verified, and it is interesting to verify them, 
by means of the approximate values of the functions: thus, for one of the equations, 
we have 
c.-AsAqA^, i.e. 
C 0 C 12 A.,A 15 
+ C^Cg 
= 0, 
(2 A + 2A 7 ) (- 2 A + 2A 7 ) 
-1.1. 
+ 1.1. 
= 0, 
1 . 1 
2 A cos ^7r (u + v) + 2A 7 cos t (u — v). 
— 2A cos r (u + v) + 2A 7 cos (u — v) 
2 A sin 2"7r {u +v)— 2A' sin \ir (u — v). 
— 2 A sin ^7r (u + v) + 2 A' sin r (u — v) ; 
2Q. 1.2Q cos \mu . 1 
2Q. 1.2Q cos tu. 1 
viz. the equation to be verified is here 
- 4A 2 + 4A' 2 
+ 4A 2 cos 2 -^-7r (a + v) — 4A' 2 cos 2 r (u — v) 
+ 4A 2 sin 2 \tt (u + v) — 4A' 2 sin 2 \tt (u — v) 
= 0, 
which is right. 
119. In the equation 
c\)C]-jA] Ai, i.e., 
r +4 Ag Aj2 
+ c 3 c 6 A 14 A u 
= 0, =0; 
this is right, but there is no verification as to the term c 3 c 6 A ]4 A n ; taking the more 
approximate values, the term in question taken negatively, that is, — c 3 c e A 14 A u is 
= — (2A + 2A 7 ). 2$. — 2S sin tv. — 2A sin \nr (u + v) + 2A 7 sin (u — v), 
which is 
= - 8S 2 (A + A') 2 cos |-7tu + 8S 2 (A + A') A cos r (u + 2v) + 8S' 2 (A + A 7 ) A 7 cos ^7r (u — 2v), 
and this ought therefore to be the value of the first two terms, that is, of 
(2Q + 2Q 9 — 2A — 2A 7 ) (1 — 2Q i - 2$ 4 ) [2Q cos \ntu + 2Q a cos f7tu 
+ 2 A cos §7r (u + 2v) + 2 A' cos ^7r (u — 2v)} (1 — 2Q 4 cos itu + 2S 4 cos itv) 
— (2Q + 2Q 9 + 2A + 2A') (1 — 2Q 4 + 2S 4 ) [2Q cos ^tu + 2Q 9 cos §7m 
— 2A cos -^7r (u + 2v) — 2A 7 cos \nr (u — 2v)) (1 — 2Q 4 cos mi — 2S* cos m), 
which to the proper degree of approximation is 
= (2Q — 4Q 3 — 4QS* + 2Q 9 —2A — 2A') {2Q cos \ntu — 4Q 5 cos \mi cos mi 
+ 4QS 4, cos \mu cos m + 2Q 9 cos \mu + 2A cos ^-7r (u + 2v) + 2A' cos ¿7r (u — 2v)} 
— (2Q — 4Q 5 + 4Q$ 4 + 2Q 9 + 2A + 2A') {2Q cos ¿ttw — 4Q 3 cos |7m cos mu 
— 4Q$ 4 cos tu cos m + 2Q 9 cos \mi — 2A cos \m (u + 2v) — 2A' cos |7r (u — 2v)). 
69—2 
mm