614 
PROFESSOR SYLVESTER ON THE REAL 
Hence since T to a constant factor pres is identical with C, the coefficients of j? 3 and £ 3 in 
the above determinant must vanish in order that gj? may be contained in T. 
Hence the two determinants 
a (3 y j3 y c5 
(3 y b and yd g 
y ci £ & g / 
both vanish. 
Hence either a, /3, y, or otherwise y, &, g, or else the first minors of 
(5 y 
y & 
§ g 
are each zero. 
The first two suppositions must be excluded, since either of them would lead to the 
conclusion of T, and therefore C, being a perfect cube, contrary to hypothesis. The last 
supposition implies either that j3, y, &, or otherwise that y, &, g, or else that (3^— y 2 and 
ye—h 2 are each zero. 
If (3, y, § are each zero, T becomes a multiple of jj 2 ! ; if y, &, s are each zero, T becomes 
a multiple of ; that is to say, T, and consequently C, contains a square factor ; and 
obviously the converse is true, so that when C contains a square factor F is reducible to 
y 2 8 2 y 3 
the form au & -\-5euv*-\-fv 5 . When this is not the case g=— = ^* Hence 
which is of the form <y 5 +® 5 +\|/ 5 , <p, 4 1 being linear functions of x, y. 
(33) We have supposed C not to be a perfect cube. When it is a perfect cube, say 
| 3 , we may assume q any second linear function of x, y ; and expressing F in the same 
manner as before in terms of |, 77 , it is clear that all the first minors of 
a (3 y d 
(3 y & s 
y § g /, 
except the one obtained by cancelling the last column in the above matrix, must vanish, 
consequently cS, g, i must all vanish, so that O, and consequently F, must contain a cube 
factor identical with the canonizant itself. 
Lastly, if the canonizant vanish entirely, every first minor in the above matrix, when 
we write again a, b, c, d , e, i in lieu of a, j3, y, d, g, /, will be zero. Hence either 
a , b, c , d, or b, c, d, e , or c, d , e, i must each vanish, or else that must be the case with 
the first minors of 
abed 
b c d e, 
