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

ALTERNANTS (BELLAVITIS, 1857)

181

BELLAVITIS, G. (1857, June).
[Sposizione elementare della teorica dei deterninanti. Memorie
[stituto Veneto..., vii. pp. 67-144.]
Bellavitis reaches the subject of the difference-product first in
7 and then in ~ 47 of his exposition, and his proof of the
results dealt with in the preceding year by Prouhet is his
own and interesting. Denoting the equation whose roots are
a1, a2,.., at by
X~n _.pIXn-l 1 92Xn -2.. 0
and the difference-product of the roots by iH, he multiplies both
sides of the identity
(x- a1)(x —a2).. (x-a = X1an nP1Xi2-l~+P2cn2_.
by II; and as the result on the left-hand side is evidently the
difference-product of a1, a,2..., cy, x, he obtains
0I 1 n -I1Xl it I'b _231 IL -I
atlC(,... a,, x' = ( 1n Pxn +...)I
It only remains then to equate like powers of x and there
results
a a a2 cIa-2 Pi
1 n-3 t-i n
ala2 an-2azcc,,,cal P11
1 2 3 1-
1,a2%.... anian1a =P2"'
He points out also that as an alternative to this we may begin
with I.. alxa 1, express it as a determinant of the next
lower order, remove the factors (x- a1), (x-a 2),...(x, -,
change the product of these into XnpXn-l +.... and then
equate coefficients of like powers of x as before.
Multiplying again by II he has of course
0- 1 2 n-1 it In n-1 112
ala3a. a,, x.1 nPixn+
* See footnote to page 163 above.