94 INTERFERENCE OF LIGHT. 



(Ill) These two equations contain all the laws of the 

 compound movement. Squaring and adding 



A* = di* + 2aiCt z cos (ai - a 2 ) + aj. 



And the amplitude of the resulting vibration is, conse- 

 quently, equal to the diagonal of a parallelogram whose sides 

 are the amplitudes of the component vibrations, and con- 

 tained angle the difference of their phases. When the am- 

 plitudes of the component vibrations are equal, or a* = a\, 



A* = 2#i 2 (1 + COS (aj - a 2 ) j = 4#! 2 cos 2 -^ ( - ; 

 whence 



. ~ Ct] Cf 2 



A = Za\ cos JT . 



Again, if we divide the former of the two equations of 

 condition by the latter, we have 



a\ sin ai + # 2 sin a.? 



tan p = ; 



i cos cti -f 6/ 2 cos a 2 



which gives the phase of the resulting vibration. This 

 equation is reducible to the form 



a\ sin (p - 01) + 2 sin (p - a 2 ) = ; 



from which it follows that the differences between the phase 

 of the resultant vibration, and the phases of the two compo- 

 nents, are represented by the angles made by the diagonal 

 of the parallelogram with the two sides. Accordingly, pa- 

 rallel vibrations are compounded by the same rules as two 

 forces meeting in a point. 



The amplitude of the resultant vibration is a max- 

 imum, and equal to the sum of the amplitudes of the compo- 

 nent vibrations, when cos (ai - a 2 ) = + 1, or 



ai 02 = 2mr. 



