The Pell equation, by Edward Everett Whitford.

THE PELL EQUATION

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de la forme x2 - ty2, t etant un nombre rationnel, positif
ou negatif; resolution en nombres entiers du systeme des
equations indeterminees
y = X2 + t(x + a)2, y2 = z2 + t(z + )2,"
Nouvelles annales de mathematiques, vol. XVII (2),
p. 419, p. 433, Paris, 1878.
HART, "A new method of solving equations of the
form x2 - Ay2 = 1," Mathematical questions from the
Educational Times, vol. XXVIII, p. 29, London, 1878.
Let
A = r2 = m,
then
x2 - r2y2 = 1 my2.
Put
x + ry = 1 - my2
and
x - ry = 1;
then
2 =~ my2 - my2
x= 2, ry= 2
Therefore
2r 2r m
Y - m' x = m.
Hence this method does not give integral solutions for
all values of A.
H. BROCARD, "Note sur le probleme de Pell," Nouvelle correspondance mathematique, vol. IV, p. 161, 193,
228, 337, Paris, 1878.
S. ROBERTS, "On forms of numbers determined by
continued fractions," Proceedings of the London Mathematical Society, vol. X, p. 29. London, 1878.
Among the many theorems in this article the most
interesting is: If t, u, is the least solution of
x2 - Ay2 = 1,