The theory of determinants in the historical order of development, by Sir Thomas Muir.

340 HISTORY OF THE THEORY OF DETERMINANTS
It is not noted by the author that having n +2 equations linear
and homogeneous in the A's he could at once deduce
1 N N2.... xn+1
a a, CL2... CLtn~1
a, Ca2 C3.... a,,4 = 0.
aCC,, C,+.... CCt
Cit abn~ Ctn+2 t2n+1
The other forms of the equation, however, he gives full attention to.
SYLVESTER, J. J. (1853, June).
[On a theory of the syzygetic relations of two rational integral
functions...... Philos. Transac. B. Soc. (London). cxliii.
pp. 407-548; or Collected Math. Papers, i. pp. 429-586.]
When dealing in art. 7 with Bezout's condensed eliminant of
two equations of the flth degree, Sylvester illustrates by the case
of ni =5', that is to say, where the equations are
aox5 ~ aC1x4 + a 2x3 + a3x2 ~ a4X ~ ~a = 0
b x5 + b c 4 + b cv3 ~ b3x2 + bx b 0
pointing out that the eliminant may be constructed by first
forming the array
Ia0b,1 I I a Lb 2 1 a0b3 1 b4 ctab5 I aCb
I a0b21 I a b3l I a0b, I I a,,b5 I ab, I
I a0b3j I a b b5 I aCb,51 I a,1bj I 1a2b5.
I CLa0b 4 c I aL I I aLb5l I a2b5 I ab5
I a0b5 I I alb.1 I a2b5 I cL3b5 I I a4b, I
and then, as it were, superposing the array
I alb2 I a,b 3 1 ac,b 4
C1 ba I a abb4 I aLb~
and next the array
I a2 b3 i