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

HESSIANS (HESSE AND JACOBI, 1849)

385

and thence
iA = (m-l){U 11{ + U i22 +.. + Uin,}. (a)
Differentiating both sides of this with respect to x,, we have
-i =.= (% _ 1 ) (UilUkl + Ui2k2 +. + Uinhn)
+ ( -1 )?ti + U2 3il + + in
- (n-l){qtl ++ 2 z + ' + } v
if k be"different from i. A second differentiation, but this time
with respect to x,, gives
__ ( 3_U 3DU 2U }
But on the supposition that I is different from i we have
'tlUli + 'a2Ui + Uin = 0,
and therefore by differentiation with respect to xk
aUil + Ui2 +Ui,,
UZ1 -+^ ' 2 + + + n DX,
+ 'itkllUil + 'kl2Ui2 + Z.. + knUi = 0.
Consequently by substitution
'za U ~a2Ui aUi a2Ui}
xi = (r -1) U{ + 2XaX + +. T *
- (n -1) {U7lNU1l + kZ2Ui2 + *. * + 2UzU (y)
where I and k are each different from i.
If now special values of xi, x,... x. make,m,,t...., un all
vanish, then, Jacobi says, we shall also have
A = 0, = 0, U,,s = NXs
M.D.xII. 2B
M.D). I. 2B