AND GENERA OE TERNARY QUADRATIC FORMS. 
I o 
Let (a be any prime divisor of M, and let us determine the first set of characters by 
the equations 
GMir)- 
the second set by the equations 
(9-g 
(24) 
being a particular character of f, of which the value is assigned in the proposed 
generic character. With respect to the supplementary characters of <p, it will be found 
on a comparison of the above Table with Table II. A, that, when the proposed generic 
character includes no simultaneous character, the supplementary characters attribu- 
table to <p are the same as those attributable to f; we then assign to the supple- 
mentary characters of <p the same values which are assigned to the supplementary 
characters of f in the proposed generic character. But when the proposed generic 
character includes a simultaneous character, there is always a supplementary character 
(and only one) attributable to <p, and not to f; this character of <p we determine so that 
the value of the simultaneous character of f and F, and the value of the unit similarly 
formed with m and M, may be coincident. This determination is always possible, as 
will be seen on a comparison of the cases (S) of Table II. A, with the above Table of 
supplementary characters of binary forms. As we have now assigned a value to every 
particular character attributable to <p, it is necessary to inquire whether a form <p, 
possessing such a complete character, actually exists ; i. e. whether the character that 
we have assigned to <p satisfies the condition of possibility for binary forms of determi- 
nant — DM. 
-1, according as Q, is of the form £1,0*, or 20 i 02 , that 
If, as in art. 8, +1 
condition is 
- 1 ) 
<0 — 1 (ft 2 - I 
2 u — 
(«%)- 
(25) 
A, + l 0,M+1 A,(p+1 <p 2 -l / m. \ 
(-!)—• , -i—i" .... (26) 
lint (^") = (^")> by the equations (24), and if (again as in art. 8) (J = + l. or — 1, 
according as A is of the form A^, or 2AtA2, 
(i) = (^=(-iF^x^ X (D. 
Substituting these values in (26), and observing that in every case 
n,M+i A,ip+i (p^-i m 2 — 1 f--i r 2 -i 
( — 1) 2 • a « 8 0 s 8 (3 8 , 
we obtain 
1T y S'-' , *-' { f \ / ^ \ f \ n '— • A|+1 
