160 Proceedings of the Koyal Society of Edinburgh. [Sess. 
1840. His rule of formation, which is different from Cauchy’s, may be 
paraphrased in the case of the quartics 
atffi + a p? + a 2 x 2 + a 3 x + a 4 = 0 ] 
b 0 # 4 + b pc? 4 - bpc 2 + b z x + b 4 = 0 j 
by saying that the resultant is got by superposing on the central two-line 
minor of the symmetric (jper symmetric, rather) determinant 
I I I I I I I «<A I 
\ a A\ l«Alj{ a o 6 4 l \ a A\ 
\ I I a A I I I I «2 & 4 I 
KVI | a, 6 4 1 \a 2 b\ \a 3 b±\ 
the determinant 
l«AI l«A! I 
I ° A I I I I > 
the latter being obtained from the array 
a i 
b 1 b 2 b s 
in the same manner as the former is obtained from the array 
a 0 ^3 ^4 
& 0 b-^ b^ b 3 & 4 . 
In his “ Theorie Generale de 1’ Elimination ” the matter is gone into in greater 
detail, the rule of formation occupying a full page (pp. 55-56), 
Cayley (1856, March). 
[Note upon a result of elimination. Philos. Magazine, (4) xi. pp. 
378-379: or Collected Math. Papers, iii. pp. 214-215.] 
If the quadric 
ax 2 + by 2 + cz 2 + 2 fyz + 2 gzx + 2hxy 
have a linear factor, £to + rjy + tz say, it must vanish identically when we 
make the substitution 
» j y » -z = PC~yy,y ay - , 
where a , /3 , y are any quantities whatever ; and consequently the co- 
factors of a 2 , /3 2 , . . .in the result must vanish, — that is to say, we must 
have 
