170 
MR. W. M. HICKS ON THE STEADY MOTION AND 
HI 
2a/3 m V' m tl F^ 7 Z[ » _1 
' 1 ‘^ +1 F r F,_ 1 
1 KK-j 
V _ ± _ 
2a/9 M P' M Z<1,1+1 P'r?'r-1 X l _ 1 
-v/s ’ -_1_ ^FX-! 
1 p'.p;-! 
(?i > m) 
r 
(n~m) 
( 6 ) 
It remains to show that with these values of A,„ B ;i the series 2A /; P^ and 2B ;i P„ are 
convergent. The parts of A n , B ; , depending on n, when n is large, 
* 
1 
= \t„ 
1 
Now 
*#+I p/ p/ /v ^Hlp/ p/ 
-L y-A A r ± r _i 
Ph_! P n OP n—\ 1 P?i CP},_| 
P'„ CP M Pa—, C F-P^/C 
[T.F., 12, 13] 
< c since P„>CP«_ 1 
Hence 
® 1 1 
T~l/ -p/ 
l+H+^+ • • • 
’« + ]p/ p/ ^ p/ p/ 1 I p I p 
A ;X ;._2 A W+ ^A V_y 
C 1 
< n 
n _i p> p/ 
x x ^ 2 +]^x ri 
Therefore 
A,]^ . u . 1 AC P„ 
P„ is ultimately <- 
-t> H | 
f!_i p' p' 
or the series are convergent. 
We are now in a position to determine <£ for any normal motion. All we have to 
do is to expand {S/(C —c)}*f(y) in a series of sines and cosines of multiples of v and 
consider each term as giving rise to a value of c/>, whose form we have just determined, 
and take the sum of the various values. 
A form for bxji/bu analogous to that for b<f>jbu can easily be found. If 
1 
'' WC-c) 
b-yjr 1 
- 2AJEP cos nv 
</R, 
j_S\ (C— c) ^SR„ \ A„ cos nv 
bit (C — cf 
=$ {( 2C ~~*) cos nv ~ ( cos w + 1 ^ + cos n-lv) ~ j A h 
S 
'2(0 -cf 
1 _ V . . (YFj-n . CPt„-i 
du 
-P 0 P 1 -tA 1 P 1 +^ 1 j(2C^-SB,)A„-A, +1 ^f- 1 -A w _ 1 
du 
cos nv 
