from a Vibrating Body to a surrounding Gas. 405 



slowly die away, in consequence partly of the imperfect elasticity 

 of the sphere, partly of communication of motion to the gas ; but 

 for our present purpose this need not be taken into account. 

 Moreover there will in general be a series of periodic disturbances 

 coexisting, corresponding to different periodic times ; but these 

 may be considered separately. We might therefore assume 



V= U sin mat + U' cos mat ; 



but it will materially shorten the investigation to employ an 

 imaginary exponential instead of circular functions. If we take 



Y = Ve imat , (5) 



where i is an abbreviation for */ — 1, and determine <j> by the 

 conditions of the problem, the real and imaginary parts of <f> and 

 V must satisfy all those conditions separately ; and therefore we 

 may take the real parts alone, or the coefficients of i or s/ — 1 

 in the imaginary parts, or any linear combination of these even 

 after having changed the arbitrary constants which enter into the 

 expression of the motion of the sphere, as the solution of the 

 problem, according to the way in which we conceive the given 

 quantity V expressed. 



The function cf> will be periodic in a similar manner to V, so 

 that we may take 



cf) = ylre imai (6) 



As regards the periodicity merely, we might have had a term 

 involving e~ imat as well as that written above; but it will be 

 readily seen that in order to satisfy the equation of condition (4) 

 the sign of the index of the exponential in cf> must be the same 

 as in V. 



On substituting in (2) the expression for cp given by (6), the 

 factor involving t will divide, and we shall get for the determina- 

 tion of ^ a partial differential equation free from /. Now -yjr may 

 be expanded in a series of Laplace's Functions so that 



^ = ^0 + ^1+^2+ ( 7 ) 



Substituting in the differential equation just mentioned, taking- 

 account of the fundamental equation 



1 d ( . a d^ n \ \ d 2 yjr n , 



and equating to zero the sum of the Laplace^s Functions of the 

 same order, we find 



This equation belongs to a known integrable form. The integral 



