358 
PROFESSOR A. R. FORSYTH ON 
X—a 1 = (<% — a-^.sr 1 
x—a 2 = — (a 3 —« 1 )c 3 
x —« 3 = — (« 3 — «j ) d z 
and so 
xt _(flg—a 1 ) 2 v a 3 — Oj {scrfS 2 + SCDs 2 } 
1 (« 2 — — & 2 s 2 S 2 ) 
=Sbo\ 
\/<h—<h 
With the ordinary notation for the second elliptic integral we have 
E^v u- 3 — a^) — u\/• 
U 
a., — « n — 
and since 
this gives 
V c h~ a i 
that is 
3 1 V a z~ a \ 
u-\-v-\-w=Q 
(U+V+W)=E(« v /^=^)+E(i;v / .ct 3 —«i)+E(w\/a 3 —< 
= & 2 sScr 
\/ a t a \ 
u+y+w=-N 1 
agreeing with the case when p=l of (22). 
21. The evaluation of Nj in terms of the functions can be obtained in the general 
case as follows. 
Since x lt x. 2 , ... , x p , £ 1} ... , g p , p 1} p 2 , . . . , p p are the roots of 
My-N¥=0 
we have 
M 2 y 2 —N 2 z 2 =(.r— x^)(x — x 2 ) . . . (x —y> p ). 
In this write x=a m , where rnxll'ip; then 
— N 2 Q(a,„) = (a ra — x^){a m —x 2 ) . . . (< a m —p p ) 
L {N (a m ) V 3 = IJalJ (u)alj (v)alj(u -+ v) 
Ny 4 / -1 +N 2 a,/ -2 + . . . +N p =±l m al m (u)al m (v)al m (u+v). 
that is 
and therefore 
Hence 
W p In 7 , v 7 , v 7 , , x . w = p N 1 a w p - 1 + ... +N P 
S a/ w (?0«L(y)oL(« + v)=± - ~- TyJ—, -- 
= i r (a,„) 5n=j r {a,„) 
