from a Vibrating Body to a surrounding Gas. 415 



When n is any integer the integral as it stands becomes illusory ; 

 but the complete integral, which in this case assumes a special 

 form, is readily obtained as a limiting case of the complete inte- 

 gral for general values of n. 



The series in (16) are convergent for any value of r however 

 great, but they give us no information of what becomes of the 

 functions for very large values of r. 



When r is very large, the equation (15) becomes approximately 



the integral of which is 'f n = ~Re- imr + We imr , where R and R' 

 are constant. This suggests putting the complete integral of 

 (15) under the same form, R and R' being now functions of r, 

 which, when r is large, vary but slowly, that is, remain nearly 

 constant when r is altered by only a small multiple of A. Assu- 

 ming for R and R' series of the form Ar a + B?# + O* . . . , where 

 a, /3, <y . . . are in decreasing order algebraically, and determining 

 the indices and coefficients so as to satisfy (15), we get for an- 

 other form of the complete integral 



r nr \-» imrfi l 2 -4^ 2 , (l 2 -4rc 2 )(3 2 -4?i 2 )^ 



ifr n = C(tmr)-*e- tmr <\ 1- =-^ — + ^ — /\ a J - 



L 1 . oimr 1 ,2{pimry 



(l 2 -4rc 2 )(3 2 -4ft 2 )(5 2 -4rc 2 ) "1 



1.2.3(8i0i) 8 +---_| 



+ D(«r)-W- f (l 2 -4n*) (l*-4, 2 )(3 2 --^ 



(l 2 -4rc 2 )(3 2 -4rc 2 )(5 2 -4yi 2 ) 

 + 1.2.3(8imr) 3 + 



■}J 



(17) 



These series, though ultimately divergent, begin by converging 

 rapidly when r is large, and may be employed with great advan- 

 tage when r is large, if we confine ourselves to the converging 

 part. Moreover we have at once D = as the condition to be 

 satisfied at a great distance from the cylinder. If mc were large 

 we might employ the second form of integral in satisfying the 

 condition at the surface of the cylinder, and the problem would 

 present no further difficulty. But practically in the case of vi- 

 brating strings mc is a very small fraction, the series (16) are 

 rapidly convergent, and the series ( 1 7) cannot be employed. To 

 complete the solution of the problem, therefore, it is essential to 

 express the constants A and B in terms of C and D, or at any 

 rate to find the relation between A and B imposed by the condi- 

 tion D = 0. 



This may be effected by means of the complete integral of (15) 



