366 
PROFESSOR A. R. FORSYTH ON 
This equation holds for the values 1, 2, .. . , p of m and these p equations determine 
the p functions cd m (u-\-v) for values 1, . . . , p of m in terms of functions of u and v. 
Moreover, r is any one of the numbers 1, 2, . . . , p-fl, so that these equations can 
have a large variety of forms. We may thus consider the functions al m (u-\-v)(m<p) as 
known ; the P + 1 functions al p+r (u-\-v) are given in terms of them and therefore 
ultimately in terms of functions of u and v by the equation 
<„(«+,.)=! -S 
Treating (36) in the same manner as (38) it will yield p equations involving the 
double-suffix functions of ufv ; this system, together with the relations between 
them (to which reference has already been made), will furnish the complete solution 
of the addition theorem for these functions. 
Abelian functions of order 2. 
25. Consider the particular case of the preceding for which p= 2. We now have 
P(.r) = (x —cq) (x — a 2 ) 
Q(x) = (x-a 3 )(x- rq) (x—ad) 
R(a;) = P(a:)Q(cc) 
> 
x—a 2 
f R(«) 
x—a l 
X (tv 7 "^1 
rr dx | 
\/k(x) 
x — a 
\ 
h dx i 
. j 
Write 
Also 
Then 
rf) {x) = (x — aq) (x — x 2 ). 
(r= 1, 2, 3, 4, 5) . 
x x —a r —a r 
x* — a r —h r 
for 5=1, 2, 3, 4, 5; and 
cd r . 
1 - 2 — QK), Q(cq) 
L, k, l = V (« 3 ), P(cq), P(cq) 
l/d 3 ~ = <f>(a s ) = a,b 
1 
respectively 
(xy—x^sf l r l, 
[v/ a,a^bbb— v/bdqaaa] 
( 1 ). 
( 2 ). 
( 3 ) 
( 4 ) 
