330 Proceedings of the Royal Society of Edinburgh. [Sess. 
vanishes when n> 4, and has a non-zero value independent of s when 
n = 4 ; and the real significance of it is best grasped by noting — as is not 
a i Pi 7i s i 
a 2 P '2 y -2 ^2 
O-n fin 7 n K 
i the Annates— 
-that the determinant is 
the product 
COS Srf) 
S COS S0 
sin 8(f> 
s sin s0 
COS (s + 1)0 
(s + l)cos (s+ 1)0 
sin (s+ 1)0 
(s+ l)sin (s + 1)0 
cos (s + n - 1)0 
(s + n - 1) cos (s + n — 1)0 
sin (s + n - 1)0 
(s + n - 1) sin (s + n - 1)0 
The vanishing of it when n> 4 is then self-evident, and its value when 
n = 4 being 
I a i^2y3^4 I 
cos s<f> s cos s0 sin s0 s sin 0 
cos (s+ 1)0 (s+ 1) cos (s 4- 1)0 sin(s+l)0 (s + 1) sin (s + 1)0 
cos (s + 2)0 (s + 2) cos (s + 2)<f> sin (s + 2)0 (s + 2) sin (s + 2)0 
cos (s+ 3)0 (s+ 3) cos (s + 3)0 sin (s + 3)0 (s + 3) sin (s + 3)0 , 
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 
s cos s0 - cos 8(f> s sin s0 — sin s0 
(s + 1) cos («s + 1)0 -cos(s+l)0 (s + 1) sin(s + 1)0 - sin (s + 1 )0 
and so finding for its square a determinant whose every element is in- 
dependent of s, the element in the place i,j being in fact 
(j - i) cos (j - i)<f> . 
He does not note, however, that such a determinant is zero-axial and skew, 
and that its value is thus readily seen, by a theorem of Cayley’s, to be 
i.e. 
(cos 2 0 - 4 cos 2 20 + 3 cos 0 cos 3 0) 2 , 
( - 4 sin 4 <jf >) 2 . 
Cayley, A. (1859). 
[Note on the value of certain determinants, the terms of which are the 
squared distances of points in a plane or in space. Quart. Journ. 
of Math., iii. pp. 275-277 : or Collected Math. Papers , iv. pp. 460- 
462.] 
The five results given in the paper are more important than the title 
2 2 
would imply, being true when instead of Cayley’s elements 12, 13, . . . 
