795] 
NUMBERS. 
605 
shall be equal to the determinant D of the containing form multiplied by a square 
number; in particular, the determinants must be of the same sign. If the determ 
inants are equal, then (a8 — ¡3y) 2 = 1, that is, a8 — ¡3y = + 1. Assuming in this case 
that the transformation exists, and writing a8 — /3y = e, and writing also 
x' = ax + ¡3y, 
y'=yx + By, 
then conversely 
x = - ( Sx' — (By), — a!x' + &'y', 
y = J (- y x> + ay' ), = yV + S'y, 
suppose, where a', /3', y', 8' are integers; and we have, moreover, 
a'S'-ySV = i(«S-/3 7 ), = h =«, 
€ € 
that is, a 8' — /3'y' = +1 or —1, according as a8 —/3y is = + l or —1. The two forms 
(a, 6, c), (a', 6', c') are in this case said to be equivalent, and to be, in regard to the 
particular transformation, equivalent properly or improperly according as a8—¡By(=a8' —(3'y) 
is = +1 or = — 1. We have, therefore, as a condition for the equivalence of two forms, 
that their determinants shall be equal; but this is not a sufficient condition. It is 
to be remarked also that two forms of the same determinant may be equivalent 
properly and also improperly; there may exist a transformation for which a8 — ¡By is 
= +l, and also a transformation for which a.8 — (By is =—1. But this is only the 
case when each of the forms is improperly equivalent to itself; for instance, a form 
x 2 —Dy 2 , which remains unaltered by the change x, y, into x, — y (that is, a, ¡3, 7, 8 = 1, 
0, 0, —1, and therefore a8 — ¡3y = — 1), is a form improperly equivalent to itself. A 
form improperly equivalent to itself is said to be an ambiguous form. In what follows, 
equivalent means always properly equivalent. 
27. Forms for a given determinant—classes, &c. In the case where D, = b- — ac, 
is a square, the form (a, b, c) {x, y) 2 is a product of two rational factors; this case 
may be excluded from consideration, and we thus assume that the determinant D is 
either negative, or, being positive, that it is not a square. The forms (a, b, c) of a 
given positive or negative determinant are each of them equivalent to some one out 
of a finite number of non-equivalent forms which may be considered as representing 
so many distinct classes. For instance, every form of the determinant — 1 is equivalent to 
(1, 0, 1), that is, given any form (a, b, c) for which b 2 — ac = - 1, it is possible to find integer 
values a, /3, 7, 8, such that a8 - /3y — + 1, and (a, b, c) (ax + (By, yx + 8y) 2 = (1, 0, 1) (x, y) 2 , 
that is, = a? + y 1 . Or, to take a less simple example, every form of the determinant 
— 35 is equivalent to one of the following forms: (1, 0, 35), (5, 0, 7), (3, ±1, 12), 
(4, ±1, 18),—(2, 1, 8), (6, 1, 6); for the first six forms, the numbers a, 26, c have no 
common factor, and these are said to be properly primitive forms, or to belong to the 
properly primitive order; for the last two forms, the numbers a, b, c have no common 
factor, but, a and c being each even, the numbers a, 26, c have a common factor 2,