800 
MR, A. R. FORSYTH Otf THE THETA-FTTNOTIOHS, 
which agree with Bosenhain’s set except (ty), in which his left-hand side is equivalent 
t° c 3 2 c 8 3 , probably a misprint. It is worthy of remark that the sum or difference of 
the subscript numbers is the same for the same equation, which is also the case with 
many of the equations (3l)-(39). 
12. Eliminating & 5 3 between (38), (39) we have 
or by (lol) and (£) 
C 0 2 dA + Ci^cj 3 "b c -i 2 di2 3 
Similarly from the same equations 
+WV +c 6 3 V 
C 3 2 ^14 2= C 12^1 3 C 4 2 ^9 2 C 2 2 ^15 2 * • • • • • • (41) 
C 0 2 ^13 2= <? 15 2 '^2 2 d -C 4 ! 2 '^9 2 C 1^12 2 
20 2— r 2q 2 n 2o 2 n 2a 2 
°6 ^11 — °12 ^1 °15 ^2 ^9 
P 20 2 — ft 20 2 20 2 20 2 
"10 — °8 "2 °3 "9 °6 "12 
By (35), (36) 
and therefore by (34) 
which give the squares of the uneven functions in terms of squares of the even 
functions. 
13. F olio wing Bosenhain '(91), put 
Then 
*1*’ 
r 2 r 2 
_ C 1 °3 
r 2 0 2 
°0 °2 
«?= 
(,2 ft 2 
°6 . 
0 2 ’ 
c 0 C 2 
hence by (0) 
Kf t -\-Kf l = 1 
2- 
0 Zf, 2 
‘"a °9 
r 2 
/. 2/. 2 
c 6 °12 . 
» o?) 
«/+«'/= 1 
K 3 ■ 
CM 
QO 
CM 
CM 
1 
K 2 — 
' c 2 » 
°2 °8 
„ 2. 
/» 2/> 2 
C 1 c 9 
/> 2/» 2 
°12 . 
» (0 
« 3 S +«?=i 
Kg • 
c 0 3c 8 3 
Kg _ 
/. 2/, 2 5 
°0 °8 
K n 3 - 
r 2 
2_A_ 
KO - r. 
n 2 p 2 
6 1 6 8 
,/> 2/> 2 ri 2/i 2/> 2 
c 9 °3 ^6 °15 9. 
say: . . 
c o c s 
C 0 " C 2 C 8 
Kt—Ko 
1 
c* 
cs 
TT 2 
/t 2ft 2 ft 2 
C 0 6g 
— h 3 
ft 2 ft 2 ft 2 
_ c 9 6 18 6 15 
_Tf 2 
^ 2^ 2^ 2 
°0 °2 °8 
—-^1 
■ • («) 
• • (43) 
• • (44). 
• • ( 45 ) 
• • (46) 
• • ( 47 ) 
