SECT. II.] PARTICULAR SOLUTION. 241 



It follows from this that we may take, instead of b 1 ,b z) b 3 ,... 

 Z&amp;gt; 4 .,...6 n , the n consecutive sines which are obtained by dividing the 

 whole circumference 2?r into n equal parts. In fact, denoting the 



T7&quot; 



arc 2- by u, the quantities 



iv 



sin Qu, sin lu, sin 2w, sin 8w, ... , sin (71 1) u, 



whose number is n, belong, as it is said, to a recurring series 

 whose scale of relation has two terms, 2 cos u and 1 : so that 

 we always have the condition 



sin iu = 2 cos u sin (i l)u sin (i 2) u. 

 Take then, instead of b lt b 2&amp;gt; b B ,... b n , the quantities 



sin Ow, sin lu, sin 2w, . . . sin .( !) u, 

 and we have 



q + 2 = 2 cos u, q = 2 versin it, or ^ = 2 versin . 



Iv 



We have previously written q instead of -=, so that the value 



n/ 



2k 27T 



of ^ is -- versin ; substituting in the equations these values 

 of b t and h we have 



_2A* . 2JT 



a = sin Oue m &quot; &quot; ^ 



_ verein 



3 = sm zue &quot; &quot; 



a n = sm w 



262. The last equations furnish only a very particular solu 

 tion of the problem proposed ; for if we suppose t = we have, as 

 the initial values of a 1? 2 , a 3 , ... , the quantities 



sin OM, sin Iw, sin 2u, ... sin (n 1) M, 



which in general differ from the given values a lt a a , a a) ...a n : 

 but the foregoing solution deserves to be noticed because it ex 

 presses, as we shall see presently, a circumstance which belongs to 

 all possible cases, and represents the ultimate variations of the 



F. H. 16 



