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

LESS COMMON SPECIAL FORMS (CAYLEY, 1874) 485

CAYLEY, A.. (1874).
[Note sur une formule d'inte'gration indelfinie. Comptes Rendus...
Acad. des Sci. '(Paris), lxxviii. pp. 1624-1629; or Collected
Math. Papers, ix. pp. 500-503.]
The peculiar structure of the interesting determinant which
turns up here is best made known by giving the examples
1mp-nq 1 (m -1)p -nq 1
q M (p2 +pq), 2q (m -1) (p2+pq) (rn+1)p-(n-1)q
q2 ~~~M (p2 +pq)
1 (m -2)_p- nq 1
3q (m-2)(p2 +-pq) mp-(n-1)q 2
3q2. (m-1)(p2+pq) (m-t2)p-(n-2)q
q3..M(p2 [-pq)
(m -3)p -nq1
4q (m -3) (_p +_pq) (n- 1)_p- (n -1)q 2
6q2. (in -2)(p)2 +pq) (+1I)p - (n-2)q 3
4q3 (in- l) (p2 +pq) (mt+ 3)p -'( - 3)q
q4...M (p2 + pq)
These are stated to be equal to the expressions
MP 2+nq2, M (M-1J)p4 A- 2mnp 2q2 + n(n -1)q4,
m(m-1)(m-2 )p6 + 3m(m-1)np4q2 + 3m(n-1)np2q4
-H n(n-1)(n-2)q6,
m(m-1)(mn-2)(m-3 )p8 + 4m(m-1)(m-2)np6q2 +.
-H n(n-1)(n-2)(n-3)q8,
which again are denoted by
([M]p2 + [n]q2)1, ([m]p2 -1 [n] q2)2, ([M]p2 + [n]q2)3.....
on the understanding that
[m"= m(m-1)(m-2)....(m-r+1).
Although in each determinant the complementary minors of the,
elements of the first column are calculated, it is not clear that this
was how the results were obtained.