194 
ME. W. M. HICKS ON THE STEADY MOTION AND 
To the order here reached 
a= ?7T ( “ ^ 
2 tt\ 2 0 R' 
P^*' 
0 
da 1 A, 
dk 2i r P, 
k ~ — — ak 
’.-1 
therefore 
Hence 
Now 
dU 
dk 
= -m 
-\ 
(dk 2 
n 3 
(4n 3 -l)^-2[M + U 2 M = 0 
4»CP» 
(4» s -l)f i -2=2 
- LV » \-t - 1 71—1 
■1 
= 4 n 
Therefore 
Pm+1 + P»-1 
p _ p 
x »+l x »-l 
M+ — . P,+1 - P -^ M=0 
^4 a 2 k 2 P^ + P,^ 
The coefficient of M is always positive; hence the hollow is stable for displacements 
of this kind, and the time of vibration for displacement of order n is 
Anrede 
~U 
w 
1P»+1 + P>1-1 
(l PK+l p»—1 
Since throughout our approximations we have neglected kr compared with unity, we 
may simplify this further by obtaining the value of the expression under the square 
root to the same order, 
Now 
(2 n + 1)(P M+1 + Pn-d 4/tCP „— 2 P 
(2n + 1)(P» +1 - Ib_ 1 ) _ 4nCP„ -AnP,^ 
1 (2»-l)P,_ 1 
2 nC' (2n — l )P„ 
!_1 (2n —1)P, ! _ 1 
C' (2w-l)P» 
2n-l 
8n(n— 1)C 2 
2k-1 
4(?i —1)C 2 
i _ 
2n(n— 1) 
n — 1 
= 1+ ^F 
2n[n—1 ) 
