1896-97.] Dr T. Muir on Ternary Quadrics . 
221 
raise the presumption that Sylvester had not hit upon the simplest 
mode of performing the elimination. 
(2) Looking to the terms which do not contain x in the first 
two equations, we see that by multiplying both sides of the first 
equation by My + lz , and both sides of the second by az, we shall 
have, after subtraction, an equation consisting of terms all contain- 
ing x, and from which this factor may be removed. The result in 
fact is 
cMy 2 + (bl — am)z 2 + (6M + cl - an)yz + A Izx + AM xy = 0 
—a derived equation which is already simpler than Sylvester’s, 
but which may be further simplified by using the second of the 
original equations to eliminate the term containing y 1 . Doing 
this we have 
(bl - am)z 2 + (6M - an)yz + (A l - cm)zx + (AM - cn)xy = 0. 
In a similar manner it may be shown that 
(nq - mr)x 2 + (MR - lp)yz + (AE£ - mp)zx + (M q - lr)xy = 0 , 
and (pc - ra)y 2 + (Rc - qa)yz + (RA - qb)zx + (pA - rb)xy = 0 ; 
but, having obtained one equation, we may obtain the two others 
of like kind by merely changing the letters in accordance with 
the cycles 
this being the mode in which any one of the original set is obtained 
from one of the remaining two. 
(3) Another mode of obtaining the equations of the preceding 
paragraph may be noted in passing. It is much simpler, although 
less likely to occur to one at first. Writing the first two equations 
of the original set in the form 
(Ax + bz)x + (az + cx)y = 0 | 
(ny + mz)x + (My + lz)y — 0 J 
and eliminating dialytically we at once obtain 
