798 
HR. A. R. FORSYTH ON THE THETA-FUNCTIONS, 
3c 0 3 V = 
3Cg 2 ^2 = 
^ C 15 3 ^10 2 = 
3tf V= 
3c 0 2 «9' 13 2 = 
3c 8 2 ^ 2 = 
^ c 6 3 '^11 3== 
3c 3 2 V= 
3ci 2 V= 
3 Cl2 2 ^ 2 = 
3c 2 2 «9- 13 3 = 
3c 3 3 V= 
^ C 15 3 ^14 3= 
^ c 12 3 "^13 3 = 
^ C 1o 3 ^11 3= 
3c 9 2 -9- 13 3 = 
c 4 3 ^ 3 + C 6 3 ^ 3 3 - Ci 3 ^ 4 3 _c W _ cW +ClW - Ci5 2 V-« 9 3 V + tf V 
C 4 2 ^ 2 + C 6 2 -« 1 2 3 V H-«15 3 V+C9 3 V-« 8 3 V 
^3 3 -^4 3 -%^5 3 + ^6 3 + VV “ C A 3 + V V +V 
% 2 ^ 2 -C 6 2 ^_ cW+cW + cW _ cW +% 2 V-^ V 
+ W + V +«3 3 V-^ V 
^V=^W-*W +o.W+^V-^i 3 V+«o 3 V-«3 3 
% 3 ^l 3 -% 3 V + ^4 3 + ^ S 3 -^9 3 -^l 3 V~V V+tf V+tf V 
^l 3 + ^5 3 V+C 9 3 ^ 3 -^^5 3 -«4 3 V + « 6 3 V-^ V+V V -% 3 V 
-V V~¥ V+% 3 V 
+c 1b 3 4 2 H s ^ 3 +V^+ V+tf V+tf V 
c is“"^0 3 ~t _c 12 3 '^’3 3 C 9""^6 3 “t C 8 3 "^7" ”1“ c 6 2 "^9 2 C 4 3 "^ll 3 c 3 3 "^Ig 2 + Gj 3 •3’ 14 3 — C Q 2 # 15 2 
^+VWWV+ < W-#]i' - c 2 3 V-«1 3 V+«o 3 V 
-C , 2 V + % 3 ^+^#-5t^3 2 + ^5 2 -W -«9 3 ^ +% 3 V WV 
V-«6 3 ^ 3 -«0 3 V-^5 2 + %^6 3 + ^7 2 -% 2 V +«8 3 V + tf V 
c 4 2 +W +% 3 V-«8 2 V 
c 0 c 2 "^ 1 o=C 1 C 3 # 9 $ n + c 4 Cg -J 12 ^ 14 . 
(xi) 
(xii) 
(xiii) 
(xiY) 
(xv) 
(xvi) 
(xvii) 
(xviii) 
(xix) 
(xx) 
(xxi) 
(xxii) 
(xxiii) 
(xxiv) 
(xxv) 
(xxvi) 
(xxvii) 
"^0 "^5 ^0 ^5 —"^M?2 ^7 + ^8^13^8 ^13 ^10^15^10^15.(xxviii) 
^10^11^14^15 — ^8^9 ^12^13 + %$3 ^6 % — ^0 "h ^4 ^5. (xxix) 
10. By equations (i)-(x) we have 
<VV C 12 2 ^ia 2=: C l~^U 4“ e~ — c 2 2 '9‘;2“ + C [fd 9 3 
°0 ^0 °3 ^3 —°6 ^6 ^8 —°4 ^4 i °9 ^9 
C(/V C 15~^15~ = "1“ C 8 3r ^8 2= C l 2 ^ 1~ “1“ 0 4 *V 
(31) 
(32) 
(33) 
giving six distinct expressions for c 0 2 d 0 3 ; and six can be obtained for each of the even 
functions in this form. 
By (xi)-(xiv), 
“" C 8 S, ^13 3 "f” ^15 ^10~ ^ 
By (xv)-(xviii), 
p 2o 2 . _ p 2 0 2 —> 
^13 °8 ^5 — 
By (xv), (xvii), (xix), (xx), 
C l 2 ^12 2 +C 0 2 d- 13 2 = 
By (xxi), (xxii), and two others, 
°3 2 ^13* "h^;; 2 diy = 
C 15 2 ^2 2 '~ C 2 2 '^15'" == 
p 2o 2 _ [ p 2q 2 —— 
°4 ^9 n^°15 ^2 — 
W +<3l5 S -9o 3 = 
-C(W +c» s V • 
2o 2 . f* 2q 2 
ty 3 ^ll °6 ^11 * 
^ 2Q 2_^ 20 2 
c 12 ^1 °6 ^11 * 
C 12'^3’ • 
(34) 
(35) 
(36) 
(37) 
