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

150 HISTORY OF THE THEORY OF DETERMINANTS

and thus have
rs = i, i+1;.; i, n
F (ai, aj) - { fta,)}2. F(a,, a,).- (a,-as)'
F~a1,a1) =if'aj.).f'(as) (a1 - a)2 (a1 - aS)2
- - ~~f() F (aS, as).
What was desired has thus been fully attained;, it happens,
however, that because of the peculiar constitution of this expression for F(a,, as), it being such that
si+1.
f'(aj) f'(aj) - f(aj).f'(as)'
s =0, 1'.. i-i
we can proceed a little further and throw ID Iinto a more elegant
form. Thus, dividing the rows of
F(a,, a1). F(a2, a2) F. (a,, at,,)
by f'(al), f'(a2).. f'(a,) respectively, and thereafter the
columns by the same, we have, on putting
(rs) for F(a,, a,)/f(ar).f'(a,),
D (1 1) (22)...(an,) { f'(al) f'(a2).fa.)1
-( (1) (22)... (nn). I ~al 7 2
where
1(ii) -= -{~(iO) ~ (ii) +. (i, i -1) + (i, i +l) ~ + + (in)},
that is to say, where each diagonal element (ii) with its sign
changed is equal to the sum of all the other elements of its row
-together with the additional element. (io).
Borchardt's treatment of this peculiar axisymm etric determinant is dealt with elsewhere. (See Chapter XVI. under
Unisignants.) Meanwhile a remark incidentally made (p. 114)
-should be noted, namely, that Any axis ymnretric determinant
having the sum of every row equal to zero has all its primary
minors equal.