SECT. II.] OTHER SOLUTIONS. 243 



Independently of the two preceding solutions we could select 

 for the values of b t , b z , 6 3 , ... b n , the quantities 



sin0.2w, sinl.2i*, sin2.2w, sin3.2w, ..., sin(ft-l)2w; 

 or else 



cos0.2w, cosl.2w, cos2.2w, cos3.2w, ..., cos(?i l)2w. 



In fact, each of these series is recurrent and composed of n 

 terms ; in the scale of relation are two terms, 2 cos 2u and 1 ; 

 and if we continued the series beyond n terms, we should find n 

 others respectively equal to the n preceding. 



In general, if we denote the arcs 



2-7T 2?T 2-7T , . 2-7T 



, 1 , 2 , ..., (w 1) , &c., 



n n 1 n n 



by u lt M S , w s , ..., W B , we can take for the values of b lt 5 g , 6 3 , ... b n 

 the w quantities, 



sin Ow 4 , sin lw,., sin 2M 4 , sin 3w 4 , ..., sin (n 1) M, ; 

 or else 



cos Qu t) cos lit., cos 2ttj, cos SM,, ..., cos (?i 1) w 4 . 



The value of A corresponding to each of these series is given by the 

 equation 



i 2 & 



/^ = versm w, . 



771 



We can give n different values to i, from i = 1 to i = n. 



Substituting these values of b lf b 2 , b 3 ... b n) in the equations 

 of Art. 261, we have the differential equations of Art. 260 satisfied 

 by the following results : 



-^ versing -^rn 



tfj = sin Ott, * , or ofj = cos 



versin MJ 



j , 



- versinwj -^ 



3 = sin 2t* 4 , a = cos 2u,e 



/ i \ ~^ versin M * / t \ -^ versin w 



= sin (n 1) w 4 e , a 7i = cos (n 1) M 4 e * 



162 



