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

262 HISTORY OF THE THEORY OF DETERMINANTS
where J stands for the Jacobian of the u's with respect to the x's, and
[11] [22]... [nn]
for the adjugate of the Jacobian. Thus, when n = 3 we have
u u =j ax, axDuu u1 Du1!
J= p a x2 Dx3 xa u 3J= ax ax2
p u2 aU2 au2 aU2 au2 au2
2 2 3x3 1 PU2 X3 3x x2 p U2
Du3 3 u3 u3a Du3 u33 3 3
qu X2 3 x u3 X1 X2
or X.J = pul[lr] +pu2[2r] +qu3[3r],
and differentiating with respect to x8 there results
+ [r + p[2r] -2)
ax, ax, ax,'
+(q-p)3?[3r] )
J\ 3 3[tr] D
X = D3[r] r +(p- )J + (q-p)3 (3r])
Xr- = (q- p)[3r],
'ai = P?tr I + <^'i.a(u ).w
and therefore for the said set of values
DJ J Du3. 33
3a,. D X 3,

as IN atanl asserts.