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

470 HISTORY OF THE THEORY OF DETERMINANTS
The value of the integral is known to be
+(2 ) J(a 22-)( a2). -. (a-a,2);
and the new result obtained is a consequence of the fact that when,
by means of the equations
X2 g2 X2
XI + x2 + + 2 --
U12-a2 U.12 -a22 12 _ an2
x12 ~...+gxn2_2 X
21 2 2. 2_ X 22 n
+*- - +
'U2 a2 t /22 _.2 '- ' 2' t22 - an2
X2 X2 2 Xn 2
_ _ _ -
U 2 _ a2 /,2 2-a2 - n2 n2 -an2
the variables of the integral are changed into u%, u2,..., un the
integral itself becomes (p. 36) for the case where an = 0,
R n Rn-l.... R l
Na a
oRn Ran a1
I 2Dau- ua,,-, a.
a,-2 a
Rn R-i *
0 aa,
a,-,] az —2 la
J R J - R1 _ * * * J R
0 Ran-1, a,
where
I t= t 2/(^-1)(^~a)... (4i-a^-),
a_ a *..... *
12 = (/('-l 4a)(u- a)..( — )
R1 = \/(a^-z)... (2 -1- )(u ). _U 2. — ),
R,- /(a-U2)(a.-u).. (a_-u ).
If n be taken equal to 3 the result degenerates into the well-known
theorem of Legendre's regarding the complete elliptic integrals of
the first and second kinds.