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

228 HISTORY OF THE THEORY OF DETERMINANTS
according as r and s are equal or unequal: and (2) that if each
element of P be increased by y times the corresponding element of
Q, the resulting determinant, R say, is equal to F(x) - F(y)/laa,.
The former proposition the solver finds the more troublesome to
prove: the latter he establishes by noting that R is symmetric
with respect to x and y, and by performing on it the operation which
evaluates P and Q, namely,
row + x row_,1 +- 2 row,_2 +.
He does not, however, seem to observe that R can also be obtained
by multiplying Q,=y, by P in row-by-column fashion,
GAMBARDELLI, F. (1873).
[Sui coefficienti delle facolta analitiche, iv. Appendice. Sullo
svilippo delle funzioni isobariche. Giornale di Mat., xi.
pp. 86-89.]
Gambardelli's appendix, with its elementary account of isobaric
functions and its illustrative examples, resembles Bruno's paper of
1856. It is his fifth example that explicitly concerns us, this being
Brioschi's of 1858. He might, however, with equal appropriateness
and merely by equating two known expressions (Waring's and
Brioschi's) have given us as his second example
a 1.....
2a2 a, 1 a.... a,1 a
4 3 22 ~ ~ 1 a
3a3 a2 al 1.... = n* (-l)+e* (e —l) I ala' ' z
4a4 a3 a2 a....
and with no great additional trouble (see Glaisher 1879, infra)
a, -1....
Ia2 a1 -2 a \ ae a...
3a3 2a2 a -3.... = n 1 --
eI e2.. e..:
4a4 3a3 2a2 a...
as his third example. In this way inquiry might have been suggested
as to the existence of a relation between isobaric functions and a
certain form of recurrent.