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

328 HISTORY OF THE THEORY OF DETERMINANTS

or, by further putting w = v+, - xv,
Wo W1... WmW1 W2... Wm
Wm-1 Win.. W2m2
After finding other forms for M (x), and varying (Q 2) the mode
of finding them, Jacobi proceeds (~ 3, pp. 140-146) to deal with
N(x), first remarking, of course, that the one function is
immediately determinable from the other, because the problem
of representing Ul, m21.... by N(x)/M(x) is the same as the
problem of representing u;', ujl,.... by M(x)/N(x). Instead
of utilising this, however, he takes from the theory of "partial
fractions" the result
i —n+?n+l
N(cx) N(xj)
AX)> (xi - xlff(xi)
whence follows
i=nt+m+1
N(x) N mM (xi)
f(x) (xi - X)f'(Xi)
i=1
so that if we put
i=nz+m+l
R. for i?
for i +1_x)f'(xj)'
i=1
we have
N(x) aRo++a1Rj+... + anRm,
f(x)
From these two equations on solving for ao: a, a2 and substituting in co + alx + 2 X2
we obtain
1 x 2
M(X)= VO )VI V2
U1 V2 23
where = Ix1 x21 x32 2x4u4 1, or
(x) =coo W,
where co, v,+, - xV, X1 X21 X3X4PN (X4 - I).