1 
344 PROFESSOR A. R. FORSYTH OX 
(iii.) In a similar way il* we write 
n ( X ’ Ul ) = L(l-\ 2 x* 
dx 
J o (1 — A 2 *; 2 ) v 7 (1 — x-) (1 — Idx 1 ) 
(so that U = lj{x)=l —Axe 3 ) we shall obtain 
n(x, w 1 )-f-n(x, u 2 ) n(\, r 3 )-]-ii(a, r 4 ) 
2 v / (1-A 2 )(Z; 2 -A 3 ) 
log 
{(A — A) j + (B\/1 — V + C \/ Id —A")-} {(A + A,)" + (B\/1 —A' — C \/.k~ — A 2 ) 3 } 
{(A + A) 2 + (By+ C l/W- A 2 ) 2 } {(A — A) 2 + (B y/l — A 2 — C ^/Jd^f) 
the values of A, B, C being those which occur in the general case in (ii.). 
Let A=£sn« so as to introduce the third elliptic integral in the form used by 
Jacobi ; then 
IT(\, u) = u- 
sna 
ciiadim 
:M U > a ) 
■u+ 
v/(T4:a 2 )(Z; 2 -A 2 ) 
A n ( ? h «) 
and the form of the theorem is now obviously as follows 
n(u 1; a) + H(u. 2 , a)+Tl(u s , a) + n(w 4 , a) 
{(A + Ics'f + (VxV + Ckc')-} {(A - kdf + (B d' - C kc' f) 
_{(A -Jcsy + (B d' + Gkc'f} {( A + ks ') 2 + ( B A — C &c') 3 } _ 
= i log 
where s', c , cV stand respectively for sneq cna, dna. 
12. Example II. 
Take F m and F« as in example I., but now let 
F p =Azy -h (Bx+ C )y -f ( D)z - Gx 2 - Fa; -1, 
in effect the most general quadric relation. The eliminant X will be of the degree 8, 
and as there are seven arbitrary constants there will be only a single relation between 
the roots x lt x. 2 , . . x s , which can be expressed in the form 
-l^i 5 i c i *'i^i c i dj 1 
cpl § s 8 d 3 sp c 3 cl§ s$ I 
: 0. 
