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

ALTERNANTS (CAYLEY, 1858)

189

CAYLEY, A. (1858, Feb.).
[A fifth memoir upon quantics. Philos. Transac. R. Soc. (London),
cxlviii. pp. 429-460; or Collected Math. Papers, ii. pp. 527 -557.]
When dealing with the subject of the equivalence of two
anharmonic ratios, Cayley gives (p. 538) the result
1 a+a' aa' 2 -T 1
1 /3+/3' /3/'. v2 -v 1
1 y7+y ' wy w2 -w 1
(U -a) (-) ( - aa ) (v - a ) w - a)(w-a')
= (m-/3)(m-/')) v-) (v- ') (w-/)(w-/')
(U-y)(U —y') (v-y)(v-y') (w-y)(w-')
This, if we put a', '=, '=, 3,,becomes
(u a)2 (v-a)2 (W-a)2 1 a a2 1 U2
(q.-_/3)2 (v-_3)2 (w-f3)2 = 2 1 /3 2. 1 v2
(U_-y)2 (v-y)2 (_-y)2 1 7 72 1 W2
or, in later notation,
(u-a)2(v-/)2(W-_)2 i = 2 2(a, /3, ). (, v, w).
Cayley also gives (p. 539) the result
1 a a aa' 88' -' — s 1
1 /3 ' /3 ' t' -t' -t
1 y y' yy'/ Z -T-' -Ua 1
1 6 8' 8' vv' -v' -v I
(s-a) (s'-a') (s- ) ('-/ ( ' ) (-s'-') -) '-8')
(t-a) (t'-a') (t-/3) (t'-8') (t-y) (t'-y') (t - ) (' - ')
(n -a)('- a')) (qz- )(('- ) (-y)(L-y) (U- ')(ZC-)
(v-a)(v'-a') (v-/) (v'-/') (v-y)(v'-y') (v-8-)(v' —')
which is seen in similar fashion to include the identity
(s-a)2(t-/,)2(U- y)2(v-6)21 = 0.
These specialisations, however, are not referred to by Cayley.