PART II 
THEORY OF INVARIANTS IN NON-SYMBOLIC 
NOTATION 
15. Homogeneity of Invariants. We saw in § 11 that two 
binary quadratic forms / and f have the invariants 
d = ac — b 2 , s = ac' -\-a'c — 2hb r 
of index 2. Note that s is of the first degree in the coefficients 
a, b, c of / and also of the first degree in the coefficients of /', 
and hence is homogeneous in the coefficients of each form 
separately. The latter is also true of d, but not of the invariant 
s T2 d. 
When an invariant of two or more forms is not homogeneous 
in the coefficients of each form separately, it is a sum of invariants 
each homogeneous in the coefficients of each form separately. 
A proof may be made similar to that used in the following 
case. Grant merely that s-\-2d is an invariant of index 2 of 
the binary quadratic forms / and f. In the transformed forms 
(§ 11), the coefficients A, B, C of F are linear in a, b, c; the 
coefficients A', B', C of F' are linear in a', b', c'. By hypothesis 
AC'+A'C-2BB'+2{AC-B 2 ) = A 2 0+2d). 
The terms 2dA 2 of degree 2 in a, b, c on the right arise only 
from the part 2(AC—B 2 ) on the left. Hence d is itself an 
invariant of index 2; likewise 5 itself is an invariant. 
However, an invariant of a single form is always homo 
geneous. For example, this is the case with the above dis 
criminant d of /. We shall deduce this theorem from a more 
general one. 
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