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

LESS COMMON SPECIAL FOR~MS (CAYLEY, 1859) 467
their values are written by Cayley in the form
(1 2) = - ell gf
(13) = b'f~ c"e + d~ld ~ ellc - b'f - c'e' - d'd' - e'c' -f'b',
(14) = - bVe - c~ld - d~lc + b'e' ~ c'd/' + d'c' + e'b',
(23) - a/f- bl'e - c'ld - cl~c - e"b + aJ' + b'e,' + c'd' ~ d'c' ~ e'b' ~f'a'
(24) =a~'e + b//d + c//c + d//b - a'e' - b'd' - c'c' - d'b'- e-a
(34) - -a/a + a'a'.
The final lemma used in the verification may be formulated
thus:
If fromb the n quantities xj, x2,.. x1, and the n (n - 1) others
12, 13,...,In
23,...2n
nn
there be formed n lineo- linear functions of the two sets, namely,
A= XI' O + x2(12) + x,(13) ~..+ xn,(ln)
A= - x,.(12) + X2*O + x,(23) +..+ x,,(2n)
A= - XJ13) - X2(23) + X3*O +. + x,(3n)
=-x1(In) - x (2n) - x,~(n..+x.
then
Xlfl+X2f2+... fno'
it may be viewed as included in the identity
xi X2 X3.. Xfl
12 13... In x1
-12. 23... 2n x2
-13 -23.... 3n x3
-In - 2n -3n...
or in the statement that Any quadric whose discriminant is a
ze'ro-acxial skew determinant vanishes identically.
*When the coefficient of x,. iin~f,. is not 0 but (rr), the result of course is
Xlf1~ xf~.. + X,,f, x1() + x22 (22) +.x.2n
and in this connection it may be well to recall a step in Hermite's mode. of
effecting the automorphic transformation of a quadric (See under Orthog~onants),
N.D.II1. 2 G2