1894 - 95 .] on ProUem of Sylvester sin Elimination. 377 
The four equations thus obtained therefore are 
(ABC+PQR)i/02+ 2CQA'2a;2+ 2CRC'a:2/2- (2QRB'+4CC'A>y2=0 
2APA'?/z2+ (ABC+PQR)2a;2+ 2ARB'a;2/2- (2RPC'+4AA'B')a;y2=0 f 
2 BPC' 2 / 22 + 2 BQB' 2 ic 2 + (ABC+PQR)a; 2/2 _ (2PQA'+4BB'C')a;t/z=0 f 
-(2ABB'+4PC'A')?/22-(2BCC'+(4QA'B')zx2_(2CAA'+4RB'C')a;y24.(ABC+PQR+16A'B'C')a;2/2=0 ) , 
and the eliminant derived from them is 
ABC + PQR 2CQA' 2CRC' 2QRB' + 4CC'A' 
2APA' ABC + PQR 2ARB' 2RPC' + 4AA'B' 
2BPC' 2BQB' ABC + PQR 2PQA' + 4BB'C' 
2ABB' + 4PC'A' 2BCC' + 4QA'B' 2CAA' + 4RB'C' ABC + PQR + IGA'B'C' . 
16. The great interest attaching to this very neat form of the 
eliminant lies in the fact that the effect of making the interchange 
/A, B, C\ 
VQ, r, p/ 
is simply to change rows into columns and vice versa^ so that its 
symmetry with respect to the said interchange is evident at a 
glance. 
Still more interesting is the degenerate form obtained from it 
for Sylvester's case. After making the requisite substitution 
P, Q, R = A, B, C, the factors 2 A, 2B, 2C, 2, C, A, B can be 
struck out, and the result is 
B 
A' 
C' 
BB' + 2C'A' 
A' 
C 
B' 
CC' + 2A'B' 
C' 
B' 
A 
AA' + 2B'C' 
BB' + 2C'A' 
CC' + 2A'B' 
AA' + 2B'C' 
ABC + 8A'B'C' 
The result to be expected, however, is 
B A' C' 2 
A' C B' 
C' B' A , 
and thus we have the curious identity 
B 
A' 
C' 
BB' + 2C'A' 
A' 
C 
B' 
CC' + 2A'B' 
C' 
B' 
A 
AA' + 2B'C' 
BB' + 2C'A' 
CC' + 2A'B' 
AA' + 2B'C' 
ABC + 8A'B'C' 
B 
A! 
C' 
= 
A' 
C 
B' 
C' 
B' 
A 
2 
