J 08 Mr. Sudhansukumar Banerji on Aerial 



by means of the condition of continuity of normal motion 

 on the surfaces of the spheres. Wo see from this expression 

 that the disturbance produced at any point due to this 

 system of two vibrating spheres will be the same as that 

 due to a prescribed vibration given by 



ikct 



[A + A 2 P 2 (cos 6) + A 4 P 4 (cos 6) + . . .] e l 



on the surface of a single sphere of radius equal to that of 

 either sphere and having its centre at the origin. Now, from 

 a consideration of the nature of the motion produced in the 

 immediate neighbourhood of the two spheres, we can easily 

 ascribe approximate values to the constants A , A 2 , &c, 

 which will conform as nearly as possible to the true state 

 of affairs. As a first approximation, we can represent this 

 disturbance by U cos 26 . e ikct , which is the same as 



U[f P 2 (cos 0)-iP o (cos 6y\ e ikct , 



so that A = — ^, A 2 = J and the rest vanishes. This type of 

 vibration shows that while the two caps bounded by the 

 parallels of latitude of 45° and 135° are moving outwards 

 the intermediate zone is moving inwards and vice versa. 



Now, if we assume various values for the quantity ka 

 which will determine the wave-length for a particular pair 

 of balls, we can easily calculate the values of the intensity 

 at a great distance from the source of sound. First suppose 

 that ka=l, then since at a great distance 



ln,p — ikr 



we have 



+ jof^ (296 - 561t)P«(oos 0) + &c. J '— . 



Denoting the real and the imaginary parts of the expression 

 within the bracket by F and Gr, we have 



F = iA o -^P 2 (co S 0) + ^%P 4 (cos0) + &c, 

 G = -iA o +^P 2 (cos0)-^^P 4 (co S 0) + &c. 



