330 Proceedings of Roycd Society of Edinburgh. [sess. 
and follow Sylvester closely. There being now eight “ secondary ” 
variables, as we may call them, viz., x 2 , y 2 , z 2 , w 2 , xy, yz , zw, wx, 
we have to seek for four additional equations containing them. 
Employing Sylvester’s process, we first eliminate B from equations 
(1) and (2), with the result 
- Gxyz 2 + Ay 2 z 2 - C ifx 2 + JLyzx 2 = 0 ; 
and then L from equations (3) and (4), with the result 
- K zwx 2 + C w 2 x 2 - Aw 2 z 2 + Gwxz 2 — 0 . 
On proceeding, however, to eliminate C from these two equations, 
we find that this will cause the elimination of A also, the result in 
fact being 
- D xyzhe 2 + E yzxho 2 - ¥Lzwx 2 y 2 - Gwxzhy 2 — 0 , 
or 
T)zio - E wx + K xy - Gyz = 0 . (a). 
Here the trouble begins. In Sylvester’s case the first derived 
equation contained not only all the secondary variables of the type 
xy, hut also one of the type x 2 , the consequence being that, by 
repeating the process on a different arrangement of the original 
equations, he obtained a new derived equation, and, of course, 
similarly a third. In the case we are now dealing with this is 
impossible of attainment, the manipulation of the original equations 
in a different order leading invariably to the same result. Instead 
of being, as in Sylvester’s case, one of a set, the equation (a) is 
here unique. 
This fact comes out very clearly and conclusively in another 
way. It will be observed that any one of the four original equa- 
tions is derivable from another of them by changing the letters in 
accordance with the indications of the cycles 
A I) 
C Iv 
Consequently, if by any legitimate process an equation is deduced 
from two or more of the original equations, the process need not be 
repeated in order to obtain other like deduced equations; all that 
