Dr Muir on ProUem of Sylvester's in Elimination. 301 
so that the desired resultant comes out finally in the simple form 
the determinant in which ( A , say) is nothing more nor less than 
the discriminant of the quadric 
2. It is impossible to note a result like this without the imme- 
diate uprise, in one’s mind, of two questions, viz., (1) Is there no 
simpler way of obtaining the eliminant by means of the dialytic 
method ? (2) How comes the given problem to be connected with 
present short paper is to contribute towards the answering of these 
questions. 
3. As regards the first of them, it is clear at the outset that if 
we are to avoid a determinant of high order, like Sylvester’s, we 
must not retain in the same equation terms in £cy, yz, or zx along 
with terms in y^, or but must aim at obtaining a set of equations 
involving only one of these two triads. How, as Sylvester’s derived 
set of equations may be got from the original set by looking upon 
A, E, C as the unknowns in the latter and solving accordingly, it is 
suggested to us to write the given equations in the form 
and, as it were, solve for z^. This procedure leads to the 
equations 
A C' E' 
C' E A' =0, 
E' A' C 
Asx? -1- Ey2 + Cz^ 4- 2A'yz + 2~B'zx -i- 2C'xy . 
the finding of the discriminant of a quadric? The object of the 
This set (y) resembles Sylvester’s ; and, we may note in passing, 
from the two taken together we have the resultant 
