FROM A VIBRATING BODY TO A SURROUNDING GAS. 
461 
T'{x) denoting the derivative of the function r (n). Putting 
A — B log im— A', 
we have by equations (41) of the second paper referred to 
C=(2^)-‘{iA'+[(log2- y >--^]B;, (24) 
D=(2T)-*{A' + (log2-y)B} (25) 
i being written for y/ — 1. It is shown in that paper that these values of C, D hold 
good when the amplitude of the imaginary variable x lies between the limits 0 and t, or 
that of r (supposed to be imaginary) between the limits —A and but in crossing 
either of these limits one or other of the constants C, D is changed. In the investiga- 
tion of the present paper r is of course real. 
We have now 
A'=A— B log im=(y — log 2)B 
for the relation between A and B arising from the condition that the motion is propa- 
gated outwards from the cylinder ; and substituting in (22), we have for the value of 4 > 0 
subject to this condition 
^B( y+ iog*=)(i-=£+3S-...) 
+B(^S,-^8 ! +...) ; 
or expressed by means of the descending series, 
? — _p/ g \ I 2 l a 3 a V 
to— \2 imr) ^ | 1.8ijwr ‘ 1.2(8w) 8 1.2. 3(8 imr) 3 
123252 
(26) 
(27) 
We have from (21) 
from which the complete integral of (15) for n — 1 may be got from that for n= 0. In 
the form (17) of the integral the parts arising from differentiation of the parts contain- 
ing e~ imT and e imr respectively will contain those same exponentials, and therefore the 
complete integral of (15) for n= 1, subject to the condition that the part containing 
e imr shall disappear, will be got by differentiating the complete integral for n = 0 subject 
to that same condition. The form of the integral in the shape of a descending series 
is given at once by (17). Hence we get by differentiating (26) and (27), and at the 
same time changing the arbitrary constant by writing B ,m _1 for B, 
. BJ _ mV 2 m 4 r 4 | i 
2®~ 2 2 .4 2 “ ’ J 
•\ (mr 
m 3 r 3 
m 5 r b ) 
7 \ T " 
~2 2 .4”^ 
2 2 .4 2 .6~” j 
^,=Bi 
/ 7TI 
l 2 mr ) 
Yv-Jl -1-3 , -1. 1.8.5 -1.1.3«.5.7 
\ 1.8m-" 1 " 1.2(8mr) 2 1 .2.3(8mr) 8 
+ 
-} • • 
(28) 
(29) 
