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

216 HISTORY OF THE THEORY OF DETERMINANTS
The final expression thus obtained for 3Z,.+, is found to be equivalent to the determinant form aimed at.
The other matter dealt with is the so-called 'evaluation' of the
Z's. The result of differentiating x0 being cxc-16x, we have
exe, - XC-1z1;
and from this, in the same way,
62X0 XC-1dj+ Z1(C — 1)XC2-CX
Xc-2 fx{Zi + (C- 1)Zp 0X}
-xc-2Z,
and so forth. The result thus led up to is
Z)r = X?-C & X.
PERGOLA, E. (1863).
[Questioni 5, 6, 7. Giornale di Mat., i. pp. 63-64; i. pp. 221-222:iii. pp. 190-201; iii. pp. 94-99, 377-380.1
The interest of these problems attaches mainly to the Theory of
Integers. What belongs to the theory of determinanits is usefully
segregated by Mola in his second paper, namely, the question of
the final expansion of the recurrent
a, I... I
a2 a, 2
a,3 a2 a1 3
The nature of the 'rule' which he reaches is evident from the result
given by it in the case of the fourth order, namely,
4J alaaa, _+ aa2 + a1a3 ~ a3a1
1l(t~+1) (l-l-2)(I~3) 2(2~2) 1(1~3) 8(3~1)
alaGa2 a a2a1 - a.2aa a1
I(1I+)(~3)l 1(1~2)(1~3) 2(2+1)(2+2) 4<
i.e. a14-4- 3 aj2+ a, -a a3-6a1 -(a9-6a,
where the sets of suffixes are permutations of the partitions of 4.
The final form of expansion is that of an isobaric function, namely,
ael a962... ae
1 61262....