388 Proceedings of Royal Society of Edinburgh. 
SESS. 
Prouhet (1857, Nov.). 
[Questions 410, 411. Nouv. Annales de Math., (1) xvi. pp. 
403, 404; xvii. pp. 187-190.] 
By reason of the existence of the identity 
2 s-1 cos *a = cos sa + s cos (s- 2)a + 1) cos (s-4)a+ . . . 
it is clear that the determinant 
cos na 0 cos (n — \)a 0 cos (n - 2)a 0 . . . cos 0.a 0 
cos na Y cos (n— l)a x cos ( n - 2)a x . . . cos O.aj 
cos na 2 cos (n - l)a 2 cos (n - 2)a 2 . . . cos 0.a 2 
COS ?la n cos ( n — l)a n COS ( n - 2)a n . . . cos 0.a n 
may he transformed into 
()n — 1 
COS M a 0 
o n-2 
cos n_1 a 0 
9«-3 
COS n_2 a 0 . . 
, . COS°a 0 
2 n— 1 
COS’ 1 ^ 
e)n—2 
cos” -1 ^ 
9n-3 
COS” -2 ^ . . 
. COS 0 ^ 
2»-i 
COS n a 2 
On— 2 
cos” -1 ^ 
9n-3 
COS ,l_2 a 2 . . 
. cos 0 a 2 
c)n— 1 
COS n a n 
2 n ~ 2 
COS n-1 a n 
9«— 3 
COS n-2 a n . . 
. cos°a n 
by increasing the 1st column by multiples of the 3rd, 5th, 
7th, ... , the 2nd column by multiples of the 4th, 6th, 8th, . . . 
and so forth. In this way there is deduced the result 
^ _ 2*ra(«-i) _ j) } 
where \ is the first determinant, and D is the determinant got 
from by changing the multipliers of a 0 , , a 2 , . . . into 
indices of powers of cos a 0 , cos a 2 , cos a 2 , ... . 
Again by using the identity 
sin (s + 1 )a — - sin a{ 2 s cos s a • cos s_2 a + q • cos s “ 4 a + . . . } 
on every element of the determinant 
sin ( n 4- l)a 0 
sin na 0 . . 
sin a 0 
sin ( n + 1 ) 0 ^ 
sin na x . . 
, . sin 
sin (n + l)a n 
sin na n . . 
, . sin a n 
it is seen that the factors sin a 0 , sin a 3 , . . . . can be removed 
from the rows in order, and that the determinant so produced is 
