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

4 32 HISTORY. OF THE THEORY OF DETERMINANTS

that is, if
(x -) Bts- B -2, (2n -3) Bn2,s-1 + (n -2)(n -0 )B1,-4,,82]
+ BILs - (2s +I)B~2 {5i8 (n2+)2},,i
+ 4(qn-2)(n-3)B9z 4Ts] 7 = 0.
But this is -the, case; for, as Ca'yley shos both the cofactor of
x- n and the other similar expression following it vanish identically. The verification aimed at is thus attained.
PAINYIN, L. (1858, February).
[Sur un certain syste'me d'e6quations lin''-aires8. 'fourn. (dle,Liouville) de Math. (2), iii. pp. 41-46.].
The system of equations referred to in the title of Painvin' s
paper had presented themselves to Liouville in the course of the
research which led to his " Memoire -sur les transcendantes elliptiques.~ "(Jo urn. de, Liouville' (1)-v, pp.-441-464). Painvin's
reason for taking up the subject was his belief that one of
Liouville's results could be more simply arrived at by the use of
determinants; and in a few linies -of introduction he succeeds in
showing that the result in question can be viewed as merely the
resolution of the determinant
v a.*
n(a -1) r-1 2a.
(n -1)(a -1) r -2 3a
(n-2)(a-l) r-3..... r-n+1 na
a-i r-n
into factors.
In explaflation of the process followed the case of the fourth
order
a
3(a -1) r-1I 2a.2(a-1) r-2 3
a-i r-3