522 REPORT—1862. 
represented by the ternary quadratic form X°—2YZ; and, secondly, that 
from this representation we can always deduce a binary form, which shall 
produce by its duplication the proposed form. This solution implies a pre- 
vious investigation of the theory of ternary quadratic forms, and cannot be 
properly introduced here. 
A more elementary method, however, has been given by M. Arndt (Crelle, 
lvi. p. 72). Let D=AS’, S* representing any square dividing D; M. Arndt 
observes that the ratio of the number of actually existing genera to the 
whole number of assignable generic characters is the same for each of the 
two determinants D and A. To prove this we make use of the following sub- 
sidiary proposition :— 
“If f=(a, 6, c) be a properly primitive form of any det. D, and if 8M and 
0 be two numbers relatively prime, the necessary and sufficient condition for 
the resolubility of the congruence 
ax’ +2bay+cy*=0,mod8M..... . (A) 
is that the supplementary characters of f (if any), and the particular cha- 
racters of f (if any) which relate to uneven primes dividing both M and D, 
should coincide with the corresponding characters of 6.” - 
We may add (though this is not necessary for our present purpose), that if 
§, and §, be two values of § for each of which the congruence (A) is resoluble, 
it is resoluble for each an equal number of times. 
On reference to the Table in Art. 98, it will be seen that the particular 
characters proper to the determinant A are included among the particular 
characters proper to D. Let then (I) and (I, I’) represent any two com- 
plete generic characters for the determinants A and D, the particular cha- 
racters common to the two complete characters having the same values attri- 
buted to them in each. It may then be shown that the genus (I, I”) is or 
is not an existent genus, according as (I) is or is not existent. For (1) if 
(1, I’) be actually existent, let § be a number prime to 2D and capable of 
primitive representation by some class of that genus; the congruence w*=D, 
mod. § is therefore resoluble ; 7. e. the congruence w?=A, mod. §, is resoluble, 
so that § can be represented by a class of properly primitive forms of det. A, 
or the genus (I) is actually existent. And (2) if (1°) be an existing genus, 
let f be a form included in (I), and 6 a number prime to 2D and satisfying 
the generic character (I, I’). It appears from the subsidiary proposition 
that some number © of the linear form 8mD-+4 is capable of representation 
by f; if 6 be the greatest common divisor of the indeterminates in the repre- 
sentation of @ by f, the congruence w*=A, and consequently the congruence 
w =D, is resoluble for the modulus ee i.e. > the character of which coincides 
with the character of 6, and therefore with that of the genus (I, I’), is capa- 
ble of representation by a form of det. D, or (I, I’) is an actually existing 
genus. 
If, then, « be the number of particular characters contained in (I, I’) and 
not in (I), the numbers of actually existing genera and assignable generic 
characters for the det. D are each 2« times the corresponding numbers for the 
det. A. 
It appears from this result that it will be sufficient for our present purpose 
to consider determinants not divisible by any square. If (a, b, c) be a form 
of the principal genus of sich a determinant (we suppose that a is prime to 
D), the equation ax*+ 2bay + cy?=w? is resoluble with values of w prime to 
