3G8 
ON ABEL’S THEOREM AND ABELIAN FUNCTIONS. 
cd 1 ~(u+v)—al 1 2 (u) = a,al 2 (v){cd 1 {u-\-v)al 2 {u)cd li . 2 (u + v)-\-al. 2 (u+v)cd 1 (u)cd h2 (u)} 
— (a 1 —a 3 )al l (v){al 1 (u)al 3 (u+v)al liB (u-\-v)-\-al 1 (u-\-v)ctl 3 (u)al h 3 (u)} 
— a.,— - - cd 1 (v){cd l (u)cd 2 (u-\-v)al h2 (u-\-v)- 1 r cd 1 (u- 1 r v)al. 2 (u)cd 1 : l (u)} 
~ a Z ®3 
and when in—'2 it is 
al 2 (u-\-v)—al^{u) = a l cd 1 (v){al 1 (u)al 2 (u-\-v)cd lt2 ^u-\-v) + al 1 (u-\-v)cd 2 (u)al h2 (y)} 
— (a 2 —a 3 )al 2 (v){al 2 (u)al 3 (u-\-v)al 2t3 (u-{-v) J r al 2 (u-\-v)cd 3 (u)al 2i3 {u)} 
q, _ n 
— a, -- 5 ah(r) {al 2 {u)al i (u J r v)al h2 (u-\~v) + al 2 (u-\-v)al 1 (u)cd h2 (u)}. 
a i — a 3 
A particular case of (5) is 
(a x — a 2 )al 3 al h 2 + (a 2 —a s )a/ 1 «4 i 3 +(a 3 — a 1 )cd 2 al l! 3 = 0 . 
These three equations will suffice to determine al h2 (u-\-v), al 2iB (u-\-v), cd s l (u-\-v ); 
after which the other functions may be successively obtained from the equations 
(a 2 — a 3 )Z 4 aZ 4 aZ L4 =(a 3 — a^l 3 al 3 al lt3 ^{a^—a^l 2 al 2 al^ 2 .... (6') 
(<x 4 — a 2 )al 1 al at 4 = (a^—a 1 )al 2 al h4 + (oq—a 3 )aZ 4 aZ L3 .(5') 
(a i —a 3 )al 2 al SA =(a i —a 2 )al 2 al %i +(a 2 —a 3 )al i al 2t3 .(5') 
(a 2 —a^l 5 al 5 al hb =(a 2 —a 5 )l^al 1A +(a. 0 —a^l 2 al 2 al h2 .... (6') 
(a 4 -a 5 )a^ 4i5 =(a 1 -a 5 )a^ li5 +(ffl 4 -n 1 )aZ 5 a? 1)4 .(5') 
(« 3 - <*p) a k a k 5=(«3 “ a i) a h a k 4 + K - a h )al 3 al^ 5 .(o') 
(a 2 — a 5 )al 3 al 2 ^=(a 2 —a 3 )cd- 0 al 2t5 +(a 3 —ct 5 )al 2 cd 3t5 .(5') 
the figure at the end of each line denoting from which of the equations (5) and (6) the 
particular line has been derived. 
This case has been added and all the necessary equations have been written down 
as a justification of the statement made at the end of § 24. 
