51. 



NOTE ON THE RESOLUTION OF x n + -2 cos wet INTO FACTORS. 



x n 



[From the Cambridge Pliilosophical Society's Transactions. Vol. XL, Part 2 (1868).] 



THE relation between successive values of x m + corresponding to 

 successive integral values of m is 



1 



when m l this becomes 



y? \ x/ \ x] 



An exactly similar relation holds good between the successive values 

 of 2 cos mO, thus 



2 cos(m+l) = (2 cos 0) (2 cosm0)-2 cos (m- 1)0, 

 when ml this becomes 



2 cos 20 = (2 cos 0) (2 cos 0) - 2. 



Now let v , v lt Vz &c. v n be a series of quantities, the successive terms 

 of which are connected by the same relation as that which we have seen 



to exist between the successive values of x m + - and of 2 cos md, viz. 



x m 



