FROM A VIBRATING BODY TO A SURROUNDING GAS. 
459 
These series, though ultimately divergent, begin by converging rapidly when r is large, 
and may be employed with great advantage when r is large, if we confine ourselves to 
the converging part. Moreover we have at once D=0 as the condition to be satisfied 
at a great distance from the cylinder. If me 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 vibrating 
strings me is a very small fraction ; the series (16) are rapidly convergent, and the series 
(17) 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 condition D=0. 
This may be effected by means of the complete integral of (15) expressed in the form 
of a definite integral. For n= 0 we know that 
•4/ 0 =i 2 {E+F log (r sin 2 £)} cos (mr cos %)d% (18) 
Jo 
is a third form of the integral of (15). It is not difficult to deduce from this the inte- 
gral of (15) in a similar form for any integral value of n. Assuming 
^ n =r a ^x n dr, 
and substituting in (15), we have 
r °- +ti ^7+(2«+/3 + l )r a+fi ~ l Xn + (« 2 — n*)iP~^r ? yJLr + mVjr 3 % re ^r=0. 
Assume 
a 2 — n 2 =0, (19) 
divide the equation by r a , differentiate with respect to r, and then divide by r 3 . 
result is 
^"+( 2 « +( 3 + 1 )(/ 3 - 1 ) 
The 
This equation will be of the same form as (15) provided, 
which reduces the coefficient of the last term but one to — (a+1) 2 . In order that this 
coefficient may be increased we must choose the positive root of (19), namely n, which I 
will suppose positive. Hence 
'r~ n Xndr ( 20 ) 
gives 
d\n l d_Xn _ (»+1) a , - n 
dt 2 ' r dr r 2 + 
the same equation as that for the determination of Hence expressing in terms 
of from (20), writing n — 1 for n, and replacing by -^ n , we have 
\L —f n ~ l — - y-(»- 
dr 
l V» 
