140 
NOTE ON THE THEORY OF ELLIPTIC INTEGRALS. 
[657 
remains always the same, and if x and y are algebraically connected, y can have 
only a finite number of values: there are consequently integer values p', p" for which 
1 1 
sn + p'A) = sn (u + p" A) : or writing u—p'A for u and putting p" — p' — p, there 
is an integer value p for which sn (u + £)^4) = sn jq-U- 
Similarly there is an integer value q for which sn Jp( u + 9.B) = sn ; and we are 
at liberty to assume q — p', for if the original values are unequal, we have only in 
the place of each of them to substitute their least common multiple. 
We have thus an integer p, for which 
sn + pA) = sn 
sn ^{u ApB) = sn 
1 
M 
1 
u, 
M 
u. 
There are consequently integers m, n, r, s such that 
= iR A + nB, 
p ~ = rA+ sB, 
equations which will constitute a single relation = m, if m = s, r = n = 0; but in 
every other case will be two independent relations. In the case first referred to, the 
modulus is arbitrary and M is rational. 
But excluding this case, the equations give 
B (:mA + nB) = A (rA + sB), 
or, what is the same thing, 
rA 2 — (m — s) AB — nB 2 = 0, 
an equation which implies that the modulus has some one value out of a set of 
given values. The ratio A : B of the two periods is of necessity imaginary, and hence 
the integers m, n, r, s must be such that (m — s) 2 + nr is negative. 
The foregoing equations may be written 
(m — 
A + nB = 0, 
-4+ (*-£)*=<>, 
whence eliminating A and B we have 
(ni — 
P 
M, 
— nr = 0,