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

ALTEIRNANTS (CAYLEY, '1876)16

167

CAYLEY, A. (1876): SCOTT, R. F. (1879).
[Theorem in Trigonometry. Messenger of Math., v. p. 164.]
[Note on a theorem of Professor Cayley's. Messenger of Math.,
viii. pp. 155-157.]
By substituting the proper exponential expressions for sines and
cosines, Scott proves that
sin a cos a sin(X+a) sin(Cu~a) sin (v+a)
sin/3 cosfl sin(X+/3)sin(ux+,)sin(v+/3)
sin7 cosy/ sin(X+,y) sin c-+y) sin (v+y)
=sin (y-3) sin (y-a) sin(/3-a).sin (a+/3~y+X+MA+v),
which becomes Cayley's result when a+/3 + y + X+ ~ ' ~= 0.
In his text-book of the following year he gives the similar example
1 tan a sin 2a
1 tan 3 sin 20
1 tany7 sin 2y
=-2 - sin (y~)sin (y-a) sin (/3-a)~ sina+/+
cos a cos /3 cos y'
We may note in passing that these may with advantage be assimilated to Wolstenholme's example by changing the alternating factor
<on'the right into
21 sin -1(7 - 3) sin 1 (y-a) sin ~(3a)
*co ~(-/3) cos }ya) cos A(/3-);Scott adds four additional results, namely,
1 cos a sill a sin 3a
1 cos/3 sin 3 sin 3/3
1 cosy- sin7y sin 3y
1 cos86 sin 3 sin 36
-251I1sin '(a -/3)
{Cos -21(3a +/3 +y/+8) + Cos (a +33 +y/+8) +.
sin a cos a sin 2a cos 2a
sin/3 cos/3 sin 2/3 cos 2/3
sin7 cosy7 sin 2y cos 2y
sin(3 cos 3 siu 26 cos 26