1894-95.] T)v Mxxiv on ProUem of Sylvester s in Elimination. 375 
It is seen to be of the 12 th degree, so that the seemingly small 
increase in generality made in Sylvester’s equations by putting in 
one place P, Q, K for A, B, C raises the degree of the eliminant 
from the 3rd to the 12th. This is the more noteworthy, because 
had the full increase in generality been made — that is to say, had 
the given equations been any three ternary quadrics whatever — the 
degree of the eliminant would have been no higher. 
13. Putting now P, Q, R = A, B, C, we first notice that the last 
term - 8(ABC + PQR)(ABC - PQR)2A'B'C' disappears, and then 
that the remaining portion becomes 
IGA^B^C^ + 64A2B2C2 • A'2B'2C'2 + GIASB^C^ . A'B'C' 
- 32 ABC(ABC + 2 A'B'C')(AB • BC • B'2 + BC • CA • C'2 + CA • AB • A'2) 
+ 32A2B2C2(A . B . A'2B'2 + B • C • B'2C'2 + C • A • 
+ 16(A2B2 . B2C2 . B'4 + B2C2 . C2A2 . C'2 + C2A2 . A2B2 . A'4) . 
Of this 16A2B2C2 is a factor, and the cofactor is 
A2B2C2 + 4A'2B'2C'2 + 4 ABC • A'B'C' 
- 2(ABC + 2 A'B'C')(BB'2 + CC'2 + AA'2) 
+ 2(AB . A'2B'2 + BC • B'2C'2 + CA • C'2A'2) 
+ (B2B'4 + C2C'^ + A2A'4), 
which is easily seen to be the square of 
ABC + 2A'B'C' - AA'2 _ ^B'2 - CC'2 . 
14. But now that the more general eliminant has been calcu- 
lated, a most important property of it makes its appearance, viz., 
that it is symmetrical with respect to the interchange 
/A, B, C\ 
Vq, R, P/ . 
An examination of the original equations shows (see § 17) that this 
is as it ought to be, and the hope is raised that a determinant form 
of the eliminant may be discovered which will bear the said sym- 
metry clearly on the face of it. After many trials I have succeeded 
in obtaining this very interesting determinant form, the equations 
from which it is derived being those which involve the compound 
variables 
y ?? , zx ^ , cry2 , xyz . 
