408 NOTE ON THE RESOLUTION OF x n + \-2cosraa INTO FACTORS. [51 



X 



Also as in those cases let v = 2, but let v 1 be any quantity whatever, 

 thus we have 



&c. &c. 



Then it is evident 



(1) that v n is a definite integral function of v 1 of n dimensions, and 



that the coefficient of v" in it is unity. 



(2) that if v 1 = x + - , then v n = x n + . 



OC OC 



(3) that if v l = 2 cos6, then v n =2oosn&, 



Hence v n 2 cos na will vanish when v 1 is equal to any one of the n 

 quantities, 



2 cos a, 2 cos ( a H -- 1 , 2 cos (a + 2 ) , ...... 2 cos [a+w 1- 



\ nj \ n] \ n 



and therefore 



v n 2 cos na = [^ 2 cos a] \ v 1 2 cos ( a + - - ) ^ 2 cos (a + 2 ) ...... 



L \ ?l /JL \ TO /J 



X fl, 2 cos ( a + n 1 - 



L \ n / J 



for all values whatever of v r 

 Now, put v l = x + - ; 



OC 



.'. x n + - 2 cos na 



ic" 



= 05 H 2 cos a x H 2 cos ( a H ) x -\ 2 cos (a+2 ) 



SY* np \ Y) I \ IT \ 'D I I 

 11 \ / I I \ / | 



x a?H 2cos(a + w-l ) 



L x \ nl\ 



which is the required resolution. 



Similarly, if we put v 1 = 2 cos 6, we have 

 2 cos nd 2 cos na 



= [2 cos 2 cosal 2 cos 2 cos (a + ) 2 cos 6 2 cos ( a + 2-^"" 



\ /J V w 





X 2cos0-2cos(a+?i-l ) 









