FROM A VIBRATING BODY TO A SURROUNDING GAS. 
451 
This equation belongs to a known integrable form. The integral is 
vb n —u n e~ imr 
f »(■.+ !) 
i 1 2 . imr ' 
(ft— l)w(ft+l)(ra + 2) 
2 . 4 {imr) 2 
+u' n e imr 
ft(ft+ 1) 
2 . imr 
+ 
(ft— \)n(n + l) (ft + 2) 
2 . 4 (imr) 2 
u n and u' n being evidently Laplace’s Functions of the order n, since that is the case 
with \f/„. 
It will be convenient to take next the condition which has to be satisfied at a great 
distance from the sphere. When r is very large the series within braces may be reduced 
to their first terms 1, and we shall have 
r<p=e im(at - r) 2u n +e im(at+r) 2u , n . 
The first of these terms denotes a disturbance travelling outwards from the centre, the 
second a disturbance travelling towards the centre, the amplitude of vibration in both 
cases, for a given phase, varying inversely as the distance from the centre. In the 
problem before us there is no disturbance travelling towards the centre, and therefore 
%u' n =0, which requires that each function u' n should separately be equal to zero. We 
have therefore simply 
r^ n —u n e- imr { 1 + 
n(n+ 1) («— 1) . . . (ra + 2) 
2 . imr 
' + 
2 . 4 (imr ) 2 
+ .•• + 
1.2.3. ..2ft 
2 .4.6. ..2n[imr) 
( 8 ) 
or, if we choose to reverse the series, 
r-ty n —u n e' 
1.3.5... (27.-1) 
2 n 
(2ft — 2) 2ft 
2. 4. 6... 2ft 
H- l 1 imr + 1 .2(2n- 1)2 - ( imr )* • • + 1 Is.'" 2» ' < 9 ) 
Putting for shortness f n (r) for the multiplier of u n e~ imr on the right-hand member of 
(8) or (9), we shall have 
$=le im(at - r) 2u n f n {r). 
It remains to satisfy the equation of condition (4). Put for shortness 
i {; e ~ ,mT U r )] =~b 
so that 
F n (r) = ( 1 + imr)f n (r ) — rfjr), (10) 
and suppose U expanded in a series of Laplace’s Functions, 
U 0 + U 1 + U 2 +...; 
then substituting and equating the functions of the same order on the two sides of the 
equation, we have 
e~ imc F n (c)u n , 
and therefore 
(ii) 
This expression contains the solution of the problem, and it remains only to discuss it. 
