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

468 HISTORY OF THE THEORY OF DETERMINANTS

HIRST, T. A. (1859)
[Question 489. (A determinant which vanishes for every order
higher than the fourth.) Nouv. Annales de Math., xviii.
p. 358; (2) ix. pp. 561-563.]
Hirst's theorem is that if
ars = (a, + /3,r) cos so/ + (yr~ + 8,s) sin 5s1
then' the deterrnminant
a, a,,+,.... l,s+
ca2,s a2,s+l a...
ans a,sCl1 ",sfn - I
vanishes when n> 4, and has a non-zero valne independent of
s when n =4; and the real significance of it is best grasped
by noting-as is not done in the Annales-that the determinant
is the product
a2 132 72 62
an On Yn 6& I
cos so cos (s +l p.... cos(s + n -)
scosso (s+1)cos(s+ 1>o. (s n- l)cos(s n- 1>/I
sin so sin (s + 1).... sin (s + n - i p
s sin so (s + 1)sin (s ~- Igo.... (s + nz - 1) sin (s + n - )
The vanishing of it when n> 4 is then self-evident, and its
value when n = 4 being
I a10273 641 cos sop cos 5k sin s54 s sin so
cos(s l)4I (s+l)cos(s+1)4 sin(s +1), (s + l)sin(s+ I)
cos(s~2)o (s~2)cos(s+2)41 sin(s+2) (s~+2)sin(s~+2)4
cos(s+3)41 (s+3)cos(s~3)41 sin(s+3)0 (s~3)sin(s+3)41
we have only to show that the second determinant here is
independent of s. The solver (Lucien Bignon) does this by
multiplying the determinant by itself in the form