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

168 HISTORY OF THE THEORY OF DETERMINANTS

=25llsin '(a -i3) { fCos -I(a+ 3-y-6) + cos '(a+y-,3 —8)

1
1
1
and

+ cos~(a+6-/3 —y)};
sin a., Cos a., sin 2a, COS_ 2a,...sinnfaj Cos fcla
sifl (2 Cos a2 sin 2a2 cos 2a2.. sin fla2 Cos fla2
Sgin a2nl+1.COS Ia,+
= 4n11 i aj-a) where* j>i;
cosh a sinh a cosh 3a
cosh,3 sinhj3 cosh,303
cosh y sinh y cosh 3-y

= 4 sinh (y-j3) sinh (y-a) sinh (O-a) cosh (a + 3+ y).
The second of these four he generalized later in the same year
(Messenger of Math., viii. p. 184), the angles a, 3,y, 8 then giving
place to a,, a2, a2w, the multiples 1, 2 to 1, 2,...,n and
the evaluation to
22712-2a~1 II1 sin 1 (ak - a)k>h
SCOTT, H. F. (1879).
[Notes on determinants. Messenger of Math., viii. pp. 182-187jJ
The results given by Scott at the outset (~ 3) are those assigned
above to Bellavitis (1857) and Naegelsbach (1871). He then
passes on to double alternants, opening with Zehfuss' theorem of
1859 and a companion-theorem of his own, namely,
(n_1n... (an-/3 n) an -/31....O
where (n),. stands for n(n-1)... (n-r+1)/1 2... r, and the
suffixes attached to ~? imply that the a's are the variables in the
*We have annexed the sign-factor.