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

WRONSKIANS (PROUHET, 1852) 223
and with the help of the said theorem obtains at once
X - =d cl(log A),
where A stands for
d' (Co/>)dl(lU>l)'... -di(ug n)].
Next, by using in connection with the same sets of equations the
multipliers
_+m, _ml,..O., ~m(m+l)q2, - 0_12, 0~
and performing addition, the coefficients of x0, x1,..., x are
found to vanish, with the result
{minu! m(d$)ml} - n+ 1) *m (ml u(dp)n - 1d2O + (m-i1)! du(cdlm),
this being due to the theorem in differentiation* that the expression of m+1 terms
dr(u5m) ) +-. dr(om+-l) + m(m +1)2. dr(25-2) -..
has the values
0, mn!u(dc)m, m(im+l1).mn! u(dcm-l)d2p + (ma+ 1])! du(.(dpn)
according as r < m, =m, or ==m+l. An alternative value for
x,, is thus found, namely,
m(m+l)Y St+(m+l) a i.e. dlog [um+l(ds)'I"(0+l)];
and from the two values it follows that
A = um+l(d0)1(m`+l) x a constant,
the constant being determined to be
[+12132...(,n+l),], or 1!2!3!..m!
by considering the particular case where u = =ex. The final
result thus is
u uO dQ(uO) dm(QSO),u1 d(uQ61).... dm("Pl) = 1! 23!... m tm+l(d9p)im(n+I)
umm d(u.p.).... C.(um)
* Attributed in part to Lexell (1772) and to Arbogast (1800).