ON THE THEORY OF NUMBERS. 517 
A=ap**-), B=(ak+ bp’) p*-%, C=ak? + 2bkpy + cp*” 
whence, if y=a, (A, B, C) is properly primitive ; and is so, or not, for every 
other value of y, according as C is not, or is, divisible by p. If y=0, we have 
C=0, for p*~ [2+(F)] values of /, incongruous mod. p*; if y have any 
ie 
value intermediate between 0 and a, we have C=0, for p*-’—' values of &, 
incongruous mod. p*~’, Hence the number of properly primitive forms is 
ae? Aishhde “MG ] 
se mipmcediee ) 
and similarly if p=2 it will be found that the number of properly primitive 
forms is 2%. Hence the number N of properly primitive forms, arising from 
the application of a complete system of transformations of modulus e¢ to the 
form (a, 6, c), is eII [2 —(5)}; p denoting any uneven prime dividing e, It 
remains to determine the number of non-equivalent classes in which these N 
forms are contained. For brevity, we consider the case of a positive determi- 
nant. Let [T,, U,] represent any solution of the equation T?—DU*=1, and 
let o be the index of the least solution of that equation which is also a solution 
of T?— eDU*=1, 2. e. let o be the index of the first number in the series 
U,, U,,...which is divisible by ¢; also let (A, B, C) represent any one of the 
N ‘properly primitive forms into which (a, b,c) is transformed. The trans- 
formations of modulus e by which (a, b, c) i is changed into (A, B, C) belong to 
o different sets, the transformations of ‘the same set being equivalent by post- 
multiplication, but those of different sets not being so equivalent. For if 
| a3 B| be a transformation of (a,b, c) into (A, B, C), any other transformation 
is represented (Art. 89) by the formula 
ie T,—bU,, —cU, a, 
and these two peak eieu will or will not belong to the same set, ac~ 
U, 
, satisfying the equation 
- : - r 
cording as a unit transformation | 7 
lt i Rel ee eu: | ar, (3 
y, 6 Vv, p au,, T,+06U, Y, 0 Y 
does or does not exist. Premultiplying each side of this equation by 
| 8, —B | , we find 
—Yy a 
ee ee eT..—BU,,, —CUs| 
v,p AU, ; iy et BUe 
whence, observing that A, B, C are relatively prime, we see that A, p, r, p 
are or are not integral according as U,, is, or is not, divisible bye; a conclu- — 
sion which implies that the transformations of (a, b,c) into (A, B, C) are con- 
tained in o different sets. It thus appears that, of the N transformations, 
which applied to (a, b,c) give properly primitive forms, there are « which give 
forms equivalent to (A, B,C); 7. ¢. the number of properly primitive classes 
