AND GENERA OF TERNARY QUADRATIC FORMS. 
288 
i. e. the given primitive representation of <p by f is included among those supplied by the 
positive transformations of the forms (f) into the forms (f'). Thus the number of the 
primitive representations of the forms (<p) by the forms (f) is equal to the number of 
the positive transformations of the forms (f) into the forms (f ) : to obtain the sum of 
the weights of these representations, we consider, in particular, f one of the forms of (/’) ; 
let d be the number of its positive automorphics, so that ^ is its weight, and let s be the 
number of the forms ( f ') which are equivalent to it. Then there are ds primitive repre- 
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sentations of the forms (<p) by f; but the weight of each of these representations is ^ X “ ; 
the sum of the weights of the primitive representations of the forms (<p) byf is therefore 
Extending this conclusion to all the forms of (f), and observing that 2s is equal 
to the number of the forms (f), i. e. to 2^ x n, we find that the sum of the weights of 
the primitive representations of the forms (<p) by the forms (f) is 2' 1 x*'. 
(ii) Let M=F(T, T', T") be a given primitive representation of M by F; and let 
a , (3 
cc", (3" I 
(35) 
be a matrix, of which the constituents satisfy the equations 
a' (3 " — a"/3 , =r, u"(3-cc(3"=r, a/3'-«73=F' (36) 
All the matrices, of which the constituents satisfy these equations, are then included in 
the formula 
a , (3 
cc', (3' 
a", 13" 
xM 
(37) 
in which |w| is a square binary matrix of which the determinant is +1. Thus the 
binary forms, which are represented by f simultaneously with the given representation 
of M by F, are all equivalent to one another, and to some form of (<p) ; let p be that 
form of ( <p ) to which they are equivalent, and let us suppose (as we may do) that f is 
transformed into <p by the substitution (35). Substituting successively for | v | in the for- 
mula (37), the ” positive automorphics of <p, we obtain ” representations of <p by f : 
these representations are all different, and they include every representation of <p by f 
which is simultaneous with the given representation of M by F : the weight of each of 
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them is ^ X - ; the sum of their weights is therefore equal to or to the weight of the 
given representation of M by F. Hence the sum of the weights of all the primitive 
representations of M by the forms (F) is equal to the sum of the weights of the simul- 
taneous representations of the forms (<p) by the forms (f), or, which is the same thing, 
