51] NOTE ON THE RESOLUTION OF x n + n - 2 cos no. INTO FACTORS. 409 



Hence we see that the two equations just found are particular cases 

 of the general equation from which they have been derived, v 1 heing in 

 one case numerically not less than 2, and in the other not greater than 2. 



If either x=l or = 0, t\ becomes =2, and either of the equations gives 

 2-2 cosna = f2-2 cosal 2-2 cos(a + ~ 7r ) 2 - 2 cos (a+ 2 ) 



L \ n/JL V w/J 



x 2 2 cos (a+nl-- } \ 

 L V / J 



Similarly, if either ,r= 1 or O = TT, i\= 2, and either of the equations 

 gives 



2(-l)"-2cos>m = [-2-2cosa] -2-2cosU + --j 



I - 2 - 2 cos (a + 2 2?r )1 X I - 2 - 2 cos (a + >T^1 ) . 



\ /J \ TO /J 



52 



