ABEL’S THEOREM AND ABELIAN FUNCTIONS. 
341 
Expanding the logarithms on the right-hand side the n th term gives 
2 r B 2 »-V»- 2 , " 
2%—I (l-Aa-CzO 2 *- 1 2l 
-l 
2B 2 ” 
2 // — 1 
Cl 
Vi 
2,-2 ?iL ] -Ax + Cst) 3 *- * 1 — (1 — Aa;—Cg]) 2 ”~ r 
{(l-Aif-CV } 5 "’ 1 
4B ta ~ 1 C 
c. 
2/i — l K 
9« — 1 9v -9 9»_2 
(2w-l)(l—Ax) 2 »- 2 + — o (l-A.r) 2 »- 4 C 2 (l-A^ 2 )+ . 
1.2.3 
(-l)«4FB 2,i - 1 C 
1 (2/i — 1)(A 2 + FO 2 ) 2 "' 1 j 
1- 
2(2w-l)A 1 
2h-1 
A 2 + A 2 C 2 a; 
I /97) _1 2t> _9 97?—a 
n-1 )A 2 " -2 -( 2n -1)(2n-2) A 2 "" 3 C 2 (- 3 ' A 2 " -4 
2//-1 . . . 2/i—4 . 9 , 1\ , 
-1.2,;- A W+ ' ■ ' 
(— 1)”4A 2 B 2k-1 C 
(2n — 1)(A 2 + A 2 C 2 ) 2,i_1 
1 A 2 + A 2 C 2 
( —1)“4A 2 B 2 ’ ,-1 C (A + /AC) 2 "-(A-fAC) 2 ” 
— (2w — 1 ) j (2w — 2)A 2 "~ 3 — PC 2 ——7,—— —- A 2 ” -5 4- . . .} 
2.2zi —LA r^ 2n _i)A^-2-yb 2 C 2 2 ^~ 1 - 2 ^~ q 2 - 2to ~ 3 A g»-*+ . . 
1.2.3 
(A 2 + A 2 C 2 ) 2 " 
(— 1)“2AB 2k 1 
2/AC 
after a slight reduction and writing —1 
1 
1 
_(A - /AC) 2 ” (A + /AC) 2n _ 
Hence the whole coefficient as derived from all the terms in the expansion is 
2 k 
B 
B 3 
B 
B 3 
i [ (A—iAC) 2 1 (A—/AC) 4 ' ' * ' (A + /'AC) 2 (A + /AC) 4 
2A 
/ 
B 
B 
B 2 + (A - /AC) 2 B 2 + (A + /AC) 2 _ 
8 A 2 ABC 
(A 2 + B 2 —A 2 C 2 ) 2 + 4A 2 A 2 C 2 
and this is the value of 
E(?/ 1 )-|-E(%)4-E(m 3 ) — 
