ABEL’S THEOREM AND ABELIAN FUNCTIONS. 
3 G5 
24. Returning now to (34) and using (1 4) we have 
bal s (u + v) 
bu m 
■ (36) 
^laW«+v)-aW)}= Sp^(«){«K.)^’ l +^ + ^ , 
and from these by subtraction and noticing that 
balju + ?;) 
bu.,„ bv,„ 
• (37) 
we have 
*>W m J 
• («8). 
Now if s be different from m 
bctl s (u) l m l I \ 1 i \ 
but this no longer holds when s,m are the same since al m>m is not a recognised function. 
We proceed f 
to u m so that 
We proceed as follows to obtain ~ l i u '\ ; —differentiate both sides of (20) with respect 
OlA/ n 
cil p+r (i() 
bal p+r (u ) _ _ J L cd s (u) bale'll) | _ l m al m (n) balju) 
bu„ 
= 1 ttp-pr I ((X's) bu m J Cf nL Ctp+r t (ft’f/i) bit,, 
*=P 
where implies that the value s=m is not to be included in the summation. The 
S=1 
equation quoted above (holdiifg for all values of s from I to 2p-j“0 when substituted 
in the last gives, on division by — y - al m (u), 
■*7 i&n/) 
s=p 
aL tP+ r(u)al p+r (u) + % 
7—,al(u)al } „,(u) ) = 
1 brtl m (u) 
(tm—Ctp+r bu m 
8=1 l( tt * —“p+rWVO 
and (38) may now be written in the form 
8 — p J 
aL 2 {u)-alJ(v) — $m — ^—al s (u-\-v){al s (u)al m (v)al sm (v)—al s (v)cil m (u)at Stm (u)} 
8=1 I \Ct S ) 
—al m (u-\-v)tm al s (v)al s . m (v)al m {u) — al s {u)al Sf m (u)al m {y) 
5=1 \ a s — a p+r )r {a s ) 
~{a m —a p+> )cd m (u+v){al m (u)al p+r (v)al„ ltP+r (v) —al m (v)al p+r (u)al„ !jl>+r (u) } (39). 
