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,