430 ANALYSIS OF CURVES. 



A s sin 3 6 + B s cos 3 = \/A 3 * + B s *. sin J3 <9 + tan 



= 30-93 sin (3 + 42) 



also 



A B =-1-546 B 5 = 5-776 



2 . sin J5 + tan-'( J 



= 5-97 sin (50 + 105). 



A method of analysis which, although not quite so 

 simple, is less laborious to carry out, is the following : 



Analysis by Method of Superposition. Again, a short mathe- 

 matical summary of the principles of this method of 

 analysis will be given, followed by a detailed description 

 of the method of carrying out the calculation in practice. 

 The non-mathematical reader should be able to carry out 

 an analysis from the description and illustration of the 

 method, even if he is obliged to omit the mathematical 

 introduction. 



Lemma. Let / (a, ft) 

 = sin a + sin (a + /3) + sin (a + 2 (3) + . . . 



. . . + sin (a + n 1/3) 



= sin /3 f sin a + sin (a + /?) + sin (a + 2 #) + . . . ( 

 ~ sin p\ . . . + sin (a + n - I 6)j 



[COS (a /3) -COS (a-f \ ft) + COS (a + % ft) COSJ 



cos (a -f n - 1 S + ifl) 



. 



x 2sin ' + * sin 



. sin (a + - 1 



sini/3 



w /3 



Now if <y/3 = 0,and sin ^ does not equal 0, then / (a, Q) = 



But in the special case when 4 - = tr, ft = 2 



'Z 



and the original expression becomes 



/ (a, 8) = sin a + sin (a + 2 TT) + sin (a -)- 4-rr) + . . . 



. . . -f sin (a + n - 1 2 TT) 

 sili a -f sin a -)- sin a + . . . 

 = n sin a 



