ME. A. CATLET ON THE EESULTANT OE A SYSTEM OE TWO EQUATIONS. 705 
columns, but this is not in general the case) is 
d 
he 
(zi) 
+3 
-1 
(l3) 
-1 
if in this we omit the sign =, and in the outside line write for homogeneity ad instead 
of d, and in the outside column first substituting p for 1, 2, then write for homo- 
geneity pqr instead of we have 
2 >qr 
+3 
-1 
-1 
\iz. pgr x{-\-‘^cid — lhc)-{-g^{ — \ad), for the value of the portion in question; this is 
equivalent to 
pqr 
2^ 
ad 
+3 
-1 
be 
-1 
, or as it maybe more conveniently written, 
in which form it constitutes a part of the expression given in the sequel for the Resultant 
of the two functions in question ; and similarly the remainder of the expression is at 
once derived from the Tables (b) I. to VI. 
As a specimen of a mode of verification, it may be remarked that the Resultant 
qua invariant ought, when operated upon by the sum of the two operations, 
?>a'b^-\-2h'b^-\-cba and 2p'b^-\-gd^, 
to give a result zero. The results of the two operations are originally obtained in the 
forms in the first and second columns, and the first column, and the second column, 
with all the signs reversed, are respectively equal to the third column, and consequently 
the sum of the first and second columns vanishes, as it ought to do. 
