ANALYSIS OF CURVES. 481 



i.e., an expression of the form 



sin a 4- sin (a 4- /3) + sin (a 4 2 $) 4- . . sin (a + n- 1 /3) 



becomes equal to 



w sin a when 8 = 2 TT. 



Let the curve to be anal} r sed be written in the follow- 

 ing form 



/ W = ^i = a i sin (^ 4- a i) + 2 sin 2 (0 4- 2J + a s sin 3 

 (# 4 a,) 4- . . 



Then we may write : 



+ = y ' i= a > sin ^ + a ' + + a> - sin 2 



+ 2 + ^) + a 3 sin 3 (0 + a 3 + ^) + . . . 

 Similarly 



/ (^ + ^) - y 3 = i sin (0 4 i + -^) + % sin 2 



+ a, + ^) + a 3 sin 3 (^ + 3 + ^) 4- . . . 

 Adding these three quantities together, 



+ y-i 4- y 3 = a i s in (^ + a i) + sin ^ + a i + 4- sin 



4- a. 2 |sin 2 (0 + a,) + sin 2 (0 + a 2 

 + sin 2 + 2 + 4- 



o g ) 4- 



The expression thus consists of a number of terms of 

 the form : 



sin TO + a m 4- sin m + a m + 



4- sinme4-a m + 4- 



By the previously given lemma each of the expres- 

 sions within the brackets will be zero, unless 



. 2 TT m n 

 sin -rt = 

 z w 



i.e., unless is an integer, 



in which case the general term 



= a m n sin w (6 + a m ) 



