INTERFERENCE OF LIGHT. 93 



(110) We have hitherto confined our attention to the two 

 most important cases of coexisting vibrations those, namely, 

 in which they are in complete accordance, or are completely 

 opposed. There is little difficulty in calculating the result, 

 when the coexisting vibrations are in any intermediate stages 

 of the vibratory movement. 



The general value of the displacement of an ethereal mole- 

 cule, produced by vibratory movement, is given in the equa- 

 tion of (18). If in this equation we make, for abridgment, 



%TT 2lT 



vt = 0, and -r- x - a, 

 A A 



it becomes 



y = a sin (0 - a). 



The coefficient a, in this equation, is the amplitude of the 

 displacement, and the angle, a, is the phase. Now let two 

 parallel vibratory movements having the same period, but 

 differing in amplitude and phase, be communicated simulta- 

 neously to the molecule, and let their amplitudes be denoted 

 by i and a z , and their phases by cti and a 2 . The two dis- 

 placements are 



y l = a, sin (0 - a,) ; y 2 = 2 sin (0 - a 2 ). 



Now by the principle of the coexistence of small motions, 

 these will be added together ; and, in order that they may 

 compound a single vibration similar to the components, we 

 must have 



V = y\ + y* = A sin (9 - p). 



substituting for y\ and y 2 their values, and developing, the 

 resulting equation will be fulfilled independently of 0, pro- 

 vided that 



A sin p = #1 sin a! + a^ sin a 3 ; 

 A cos p = (ii cos ai + #3 cos a 3 . 



