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

I ECURRENTS (HESS, 1872)

9 C)?

respectively, we have from him
IP3' F 11, ' q4 F,:9~ F
21 F1 F,
21r2 V —l F2 1 3fr2 2 2rF
2Vf3 3V-2 F3 '3i3 5~12 hCr F 3
3V'-4 8f3 6V/r2 "F'4
where the coefficients of the b-,'s in the diagonal decrease by 1, and
the coefficients of any other VJ, say that in the (r, s)til place, is the
sum of the coefficients in the places (r- 1, s- i) and (r- 1, s).
SIACCI, F. (1872).
[Quistione 10. Giornale di Mat., x. p. 360; or Annali di Mat.,
(2) v. p. 304. Solution by F. Tirelli in Giornale di Mat., xii.
pp. 75-78.]
The determinants occurring here are two n-line variants of
Sjjottiswoode's recurrent
ao -1..
a, xa2. xa., x -1
a x
an x'
for aoxn ~alx"1 +.... +-a, or F (x) say, the one variant P
being got from this by the initial operation
1
col2 4- coll
a0
and being equal to F(x)/a0, and the other Q by the initial operation
x
coln+] - - coll
a.
and being equal to F(x) /an after a change of sign of all the elements.
Denoting the complementary minor of the (r, s)" element in the
one by P,.,8 and in the other by Q,., Siacci affirms (1) that

aoxP,., + a,.,,, = F(x) or 0