~ 
518 REPORT—1862. 
of det. Dé, contained in (a,6,c), a properly primitive class of det. D, is 
Nive Pg jor 4 ; a result which is in accordance with the formula of 
o o 
Dirichlet ase 103). If D be negative, we have only to put o=1, as is suffi- 
ciently apparent from the preceding proof; if, however, D=—1, o—2. 
The properly primitive classes of det. De’, into which a given properly 
primitive class (a, b,c) of det. D is transformable, are always such that, com- 
pounded with the class (e, 0,—De), they produce the class (ea, eb, ec). For 
let (a, b,c) be transformable into (A, B, C) of det. De’, and let us take a form 
of the type (A, B’e, C’e*), equivalent to (A, B, C); then (a, 5, c) and (A, B’, C’) 
are equivalent. But (e,0,—De) x (A, Bre, Ce’) =(eA, eB’, eC’), therefore also 
(e, 0,—De) x (A, B, C)=(ea, eb, ec). And conversely the classes which, com- 
pounded with (e, 0,—De), produce (ea, eb, ec) are precisely the classes into 
which (a, b,c) is transformable. Thus the properly primitive classes of det. 
De*?, which compounded with (e, 0,—De) reproduce that class itself, are no 
other than the properly primitive classes of det. De* into which (1, 0,—D) 
is transformable. And it is by this substitution of a problem of transforma- 
tion for a problem of composition that M. Lipschitz has simplified and com- 
pleted the analysis of Gauss. 
A method similar in principle is applicable to the comparison of the num- 
bers of properly and improperly primitive classes. We can first show that if 
D=1, mod. 4, the double of every properly primitive class of det. D arises 
by a transformation of modulus 2 from one, and only one, improperly primi- 
tive class of the same determinant ; viz. if (a, b,c) is a given properly primitive 
form, in which a and 4 are uneven, (2 b, 7 is improperly primitive, and is 
changed into (2a, 2b, 2c) by : : ; and, again; if (2p, q, 2r), (2p’, q’, 2r') are 
two improperly primitive forms, each of which is transformable into (2a, 24, 2c), 
these two forms are equivalent, because («,),¢) is transformable into (4p, 2q, 4r) 
and also into (4p’, 2q', 47’), while it can be shown that (a, 6, c) is transform- 
able into the double of only one improperly primitive class. Also, applying 
the system of transformations, = 
ra - ie : | , to the improperly pri- 
> 
mitive form (2p, q,2r), we obtain, if D==1, mod. 8, the double of only one 
properly primitive form: in this case therefore the numbers of properly and 
improperly primitive classes are equal. If D=5, mod. 8, we obtain the 
doubles of three properly primitive forms; and we have to decide to how 
many different classes these three forms belong. It appears from Art. 89, that 
ie [28 
y,€ : 
mitive form (a. 6,¢), all the transformations are included in the formula 
2(T,—qU,), Sate | a, B 
PU A(T, +90)! ly 6 
|T,, U,] denoting any solution of the equation T7—DU?=4. Taking the case 
of a positive determinant, and employing the same reasoning as before, we infer 
that if U, be the first of the numbers U,, U,,... which is even, these trans- 
formations are contained in ¢ different sets. But is either 1 or 3 according 
as U, is even or uneven (see Art. 96, vi.) ; the three forms will therefore re- 
present three classes or one, according as U, is even or uneven; and the 
number of properly primitive classes, in these two cases respectively, will be 
three times the number of improperly primitive classes, or equal to it. If D 
be a transformation of (2p, 7, 277) into the double of a properly pri- 
’ 
