AND GENERA OP TERNARY QUADRATIC PORMS. 
271 
P", Q', Q their values derived from the equations (15) and (18), we find that 0 2 A is the 
determinant of the matrix 
' P , i'* 4 
q", p ' , q ( 20 ) 
4 , 2 , / 
Let 
GO], G[ 2 "], QO] 
GO'], G[p'], GO ] (21) 
GO], GO], 00"] 
be the matrix reciprocal to the matrix (20); we know, from the equations (15) and (18), 
that jj/]=P", 0'] = Q', 0]=Q- Again, in the reciprocal matrices (20) and (21), we 
must have 
O']O"]-0] 2 =Ap, 
W\ - 0 "] 0 ']= a 2 ", 
0]0"]-0 2 ]=Ap', 
or substituting for O"], [?], [Y] their values, 
|>']P"-Q 2 =Ap, 
qq'-p"0"]=a ? ", 
0]P"-Q' 2 =Ay. 
Comparing these equations with the equations (16), and observing that P" is not zero, 
we infer that 
0']=P', &"]=Q", 0]=P. 
The matrix reciprocal to the matrix (20) is therefore 
and, consequently, 
OP , OQ", OQ' 
GQ", OP' , OQ 
OQ' , OQ , OP", 
A^=Q'Q"-PQ, 
( 22 ) 
A 2 '=QQ"-P'Q', 
Ap"=PP' -Q" 2 . 
These equations prove that the denominators of q, q 1 , p" are divisors of A ; i. e. that q , 
(/, p" are integral numbers, because P" is prime to A. The coefficients of the ternary 
form 
f =px? + pY +fz 2 + 2 qyz + 2 q'xz + 2 "qxy 
are therefore integral ; this form is primitive, and represents primitively the form (p, q", 
p ') ; it is also a form of the given invariants [O, A] ; for its discriminant is AO 2 , and 
the greatest common divisor of the first minors of its matrix is O ; hence its second 
invariant is A, and its first invariant either +0, or — O. But when the given invariants 
O and A are both positive, <p is a positive binary form of the negative determinant 
mdccclxvii. 2 p 
