VECTOR ANALYSIS. 81 



Now, by No. 174, 



Since 



e- s - e = aa' + e'cosfc {/S^' + yy'} - e c smb 

 Therefore 



cosO = aa' 

 and 



In like manner we find 

 sin$ = sinaaa / + %(e c + 



180. If a, )8, y and ct', ^ x , y x are reciprocals, and 



$ = aaa / + 

 and N is any whole number, 



Therefore, 



e * = e a aa > + e{ + y y } + e & c 

 cos $ = cos a aa' + cos 6 {ft/3' + yy'} c sin 6 /3y r , 



sin $ = sin a aa 4- sin 6 {/^^ + yy'} 4- c cos 6 /3y'. 

 If a and 6 are unequal, and c other than zero, we may add 



181. If a, /3, y, and a, /3', y' are reciprocals, and 



$ = aI + 

 and N is a whole number, 



Therefore 



cos $ = cos al b sin a {a/3'+ /^a'} ( J6 2 cos a+c sin a) ay', 

 sin $ = sin a I + 6 cos a {a/3' + /3a'} ( J6 2 sin a c cos a)ay'. 

 Unless 6 = 0, we may add 



182. If we suppose any dyadic $ to vary, but with the limitation 

 that all its values are homologous, we may obtain from the definitions 

 of No. 171 ,. , * ,, 



(2) 

 (3) 



(4) 



G. II. F 



