426 
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MR. SYLVESTER ON THE RESIDUES OF ^ EXPANDED 
the matrix so modified becomes 
0; 
ay\ 
aS ; 
as 
ay % 
aS ; 
as ; 
0 
aS ; 
( + )■ 
cs — ey 
\ch—dy} 
as ; 
; 
cz~ey ; 
dz — eS. 
Again, adopting the method of art. (6.), we should obtain the matrix 
0; 
7; 
s 
r; 
s; 
1 hz \ 
0 
az—hh ; 
( + ) ; 
\ch~dyj 
cs — ey 
as ; 
bz ; 
cz — ey ; 
dz—eh. 
Hence it is apparent that the secondary Bezoutians obtained by the symmetrizing 
method will differ from those obtained by the unsymmetrical method by a constant 
factor a‘-, and so in general it may readily be shown that the secondary Bezoutmns. 
by the use of the symmetrizing method, will each become affected with a constant 
irrelevant factor where (m) is the difference of the degrees of the two functions, and 
(a) the leading coefficient of the higher one of the two. When (a) is taken unity, the 
Bezoutian secondaries, as obtained by either method, will of course be identical. 
Art. (9.). There is another method* of obtaining the simplified residues to anytwo 
functions U and V of the degrees n and n-\-e respectively, which, although less elegant, 
ought not to be passed over in silence. This method consists in forming the identical 
equations (of which for greater brevity the right-hand members are suppressed). 
V=&c. 
xV=&c. 
U=&c. 
■x'’.V =&c. 
j?. U = &c. 
,2;®+‘.V=&C. 
x\\]=hc. 
j;^+2.V = &C. 
&c.=&c. 
.U=&c. 
a;S+”“'.V=&c. 
* Originally gi-en by myself in the London and Edinburgh Philosophical Magazine, as long ago as 1839 or 
1840 "nd soL%earsLbsr,ue„.ly in nnconscionsnns, of that fact, tepcoduced by my Mend CavtBV, o 
whom the method is sometimes erroneously ascribed, and who arrived at the same equations by an e„t,rel> 
different circle of reasoning. 
