40 
PROFESSOR A. E. H. LOVE ON THE INTEGRATION OF THE 
and we deduce without difficulty that, when w,, = 
[u, V, iv) = cos K ct r’‘~^Xn • ( 2/5 ~ h).( 86 ). 
Forms for f, g, h could be obtained by a similar analysis ; but the values that they 
would take near the boundaries and %\ can be written down immediately from the 
formula (80), viz. : we have, at the boundaries, 
{f, g, h) — — n(n + 1) cos k ('^^■) P«(p) • (*> ?/> 2 ) . . . (87). 
When and are nearly equal, this formula holds approximately for all values of 
r that are involved. 
33. The kinetic energy of the mode of oscillation here discussed can be calculated 
without difficulty. We require the value of 
= «« r<''- f‘V {x. (-'■) P 2^ ( 1 - 
^ njn + 1) 
2 2n + 1 
K~C- sin^KT Ct 
I {r'^^\n{Kr)Ych 
• (88). 
To calculate the Integral in this expression, we have recourse to the differential 
equation 
n (n + 1)' 
_L -2 
X. («'■)} = 0 , 
. . . . («9). 
and the condition that 
d 
dj 
. {>'”‘x»G')i = 0 
(81) 
and 
(1 - 2/it- + Jd)-i = 
liv + 
(1 - 2//i- + ' (1 - 2/it + t-)? 
00 
IF 
n 
+ (2/i - t)^t" . 
0 I 
1 + Et-(p,, + 2,/!^ - 
1 \ d/i (f./i / 
'^Pft-i ^ '''P« _ p 
dfji dfi 
and we also have the known relation 
