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

ALTERNANTS (GARBIERI, 1878)

165

where the c's and c"s are defined by the identical equations
(x-aL)(x-2). ~. (x-a,) - COxn+cX-l+. +c,,
(x-f31)(x-132).~ ~ (x-3n) - CXn+Clt- +... +C.
The mode of proof preferred is similar to that chosen in the case
of the first theorem, save that the right-hand member is now multiplied by the two difference-products
g (,(1' a2,... a,,Y, yo, I(Y,. *2 *... IO, yZ 1, *,., Zh)
in succession, and that prior to multiplication the determinant
form of each of these is raised in the usual way from the (n+h+l)th
order to the (n+2h+2)th. Another proof, however, is indicated,
dependent on the use of the first theorem; and it is also pointed
out that the second theorem includes the first.
Finally, note is taken of Borchardt's special case of 1859., where
h =-1; and the fact that Cauchy's result of 1841, though not
included, may be deduced with ease by putting
F(x, y).= x(x) — yV(y ),
x-y
and fr(x)= (x —})(x-/j)... (x-On).
CROCCHI, L. (1878).
[Sopra le funzioni Aleph ed il determinante di Cauchy. Giornale
di Mat., xvii. pp. 218-231, 380.]
After a study of Trudi's paper of 1864 regarding the resolution
of x x1 x0 x4 X, Crocchi is led to hazard the assertion that for
a similar reason any Jacobian ought to be divisible by the differenceproduct of the variables, and that the search for the quotient may
be furthered by means of the relation
| a. f2 | _ f a f2 an.| Y1.2 dayn
ox, aX2 ax DY1 2 D'yn | x1 ax2 d x,
Nothing is said regarding the f's being symmetrical with respect
to the x's. As an illustration of his suggested process for finding
the quotient he takes Trudi's case, choosing for this end
A = Sa+ (a + ), y1= sl-,
f +2 = Ys1 (/3 1), = S2= 82,
J3 = sY+l (y+l), /3 = -33,
f4 = 8+I1 -(S +1),?4 = S4-4,