AND DOUBLE THETA-FUNCTIONS. 
$ 
2.6 
0.4 
8.12 
10.14 
= 0 , 
$ 
1.5 
3.7 
11.15 
9.13 
c 
- 8.12 
0.4 
- 2.6 
c 
- 8.12 
- 0.4 
2.6 
8.12 
- 2.6 
0.4 
8.12 
2.6 
- 0.4 
- 0.4 
2.6 
- 8.12 
0.4 
- 2.6 
- 8.12 
2.6 
- 0.4 
8.12 
- 2.6 
0.4 
8.12 
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 
^3^15^0^12 
(2A+2A')(- 
2A + 2A') 
1 . 1 
C 0 Ci 2 d 3 di5 
— 1 
1 
2 A cos \tt(u-\-v)-\-2A.' cos \tt(u — v). 
— 2A cos ^tt(u-\-v)-\-2 A' cos v) 
~b {qCgdydn 
+ 1 • 
1 
— 2A sin ijr7rfy+r)— 2A' sin ^ 7 t(u — v). 
— 2A sin \tt(u-\-v) + 2A' sin \tt{u — v ) 
= 0 , 
= 0 , 
viz., the equation to be verified is here 
— 4 A 3 +4 A' 3 
+ 4A 3 cos 3 \tt (u 4- v) —4A' 3 cos 3 \tt(u — v) 
+ 4A' 3 sin 3 \tt(u-\-v) — 4A' 3 sin 3 — v) 
= 0, which is right. 
119. In the equation 
^9^12*'1 ^4 ' I ' C * 
ClC 4 3-g d 12 
+ CgC 6 d 14 d 11 
= 0 , 
2Q.1.2Q cos \ttu. 1 
— 2Q.1.2Q COS ^TTU.l 
= 0 ; 
this is right, but there is no verification as to the term c 3 c c d 14 d n ; taking the more 
approximate values, the term in question taken negatively, that is —CgC 6 d u d n is = 
“(2A^-2A , ). 2S. —2S sin \ttv. —2A sin sin t(u — v), 
which is = 
— 8S 3 (A-hA / ) 3 cos -|77i^-(-8S 3 (A-j-A / )A cos |7r(w-h2fy-b8S 3 (A-i- A y )A' cos ^tt(u — 2v) 
