516 REPORT—1862. 
place to give an account of this investigation, and to point out its relation to 
the method pursued by Gauss. We begin with the theorem 
« Every properly primitive class of determinant De? is contained in one, and 
only one, properly primitive class of determinant D.” 
Let (A,B,C) be a properly primitive form of det. De’, inwhich A is prime to ¢; 
let B! be determined by the congruence eB’=B, mod A, and C! by the equation 
12 
C' Bg ; then the forms (A, B, C) and (A, Ble, C'e*) are equivalent ; but 
(A, Ble, C'e?) is contained in (A, B’, C’), therefore also (A, B, C) is contained 
in (A, B’, C’), that is, in a properly primitive form of determinant D. Again, 
if (a, b, c), (a’, b,c) are two forms of det. D, each containing (A, B, C), these 
two forms are equivalent. For applying to (A, B, C) the system of transfor- 
m, 
mations of modulus e, included in the formula | 0 7 | (art. 88), we readily 
find that, of the resulting forms, one, and only one, will have its coefficients 
divisible by e?*; therefore the class represented by (A, B, C) contains one, 
and only one class of det. De‘, and of the type (ép, eg, @r). But, applying 
to (A, B, C) the transformations inverse to those by which (a, 6, c) and 
(a', b', c') are changed into (A, B, C), (A, B, C) is changed into (ea, €*b, e’c) 
and (¢éa', eb’, e’c’); these two forms are therefore equivalent ; 2. ¢. (a, 6, ¢) 
and (a’, 0’, c') are equivalent. 
We have next to ascertain how many different properly primitive classes of 
determinant De’ are contained in the class represented by (a, 6, c), a properly 
primitive form of det. D, in which a may be supposed prime toe. Applying to 
(a, 6, c) a complete system of transformations of modulus e, we inquire in the 
first place how many of the resulting forms are properly primitive. or this 
purpose we observe that if e=e, xe, xe, X ...(é,, @, ++. representing factors 
of which no two have any common divisor), a complete system of transforma- 
tions for the modulus ¢ is obtained by compounding, in any definite order, the 
systems of transformations for the modules ¢,, ¢,,...; te. if | e, |, | & |5-+- 
be symbols representing complete systems of transformations for the modules 
€,, &»++., every transformation of modulus ¢ is equivalent by post-multiplica- 
tion} to one and only one of the transformations | e, | x | e,| X | & | X-- 
It will, therefore, be sufficient to determine the number of properly primitive 
forms obtained by applying to a properly primitive form a complete system of 
transformations for a modulus which is the power of a prime. Let p be an 
uneven prime, and let (a, b, c) be changed into (A, B, C) by load “i 
>a 
formula which will represent a complete system of transformations for the 
modulus p”, if y receive every value from 0 to @ inclusive, and if & be the ge- 
neral term of a complete system of residues, mod p*~’ ; we find 
* em ‘i | transform (A, B, C) into (P, Q, R), we have 
P=Am?, Q=m(A‘+By), R= AX?+2Bhp+Cp2. 
Observing that A is prime to e, we infer from the congruence P=0, mod. e%, that m=e, 
p=1; the competes =0, mod. e”, then becomes A+++ B=0, mod. e, giving one, and only 
one, value of £ mod. e; and this value satisfies the remaining congruence R=0, mod. ¢, 
since AR=(Ak+B)?—De?. 
t If] A| and | B | are two transformations connected by the symbolic equation 
|B}=|A|x|V], 
in which | V | is a unit transformation, | A | and | B | are said to be equivalent by post- 
multiplication, or to belong to the same set. A complete system of transformations for any 
modulus contains one transformation belonging to each set. 
