1894 - 95 .] Dr Muir on Problem of Sylvester’s in Elimination. 381 
we have 
qx^-20'x^y 
- 2k'xyz 
A 
K 
= 0 , 
or 
x{ - 2APMyz - (ABC + PQR)a ;2 + 2 ABB'a -2 + 2V;RC'xy} = 0 , 
as it ought to be. 
Further, by making in this the interchange 
'A, B, C, X, y, z 
Q, R, P,l, - 
X y z 
we obtain 
- 2QCA':c2 _ (ABC + PQR)t/^ + 2qKB'xy + 2BCC'^a; = 0 , 
which is another equation of like form (§ 5). 
21. Not essentially different from this general method is another 
process which consists in taking the determinant of the 10th order 
connected with the above-mentioned ten equations, and transforming 
it into a determinant of lower order by adding multiples of rows 
(not columns) so as to increase the number of zero elements. For 
example, the initial form of the determinant being 
Q . . 
. R . 
. . P 
. A . 
B 
C 
-2C' 
Q 
2A 
R 
A 
-2B' 
P 
B 
-2C' 
R 
B 
-2 A' 
-2B' 
Q 
-2C' 
-2A' 
-2B' 
-QEB' -RPC' -PQA' -CAA' - ABB' -BCC' ABC + PQR + 4 A'B'C' 
we may multiply the 4th, 5th, and 6th rows by R, P, Q respectively, 
and then subtract from them C, A, B times the 1st, 2nd, 3rd rows 
respectively. This gives us 
