100 
THE HON. J. W. STRUTT ON THE THEORY OF RESONANCE. 
be able to find some motion for the inside satisfying the boundary conditions and the 
equation of continuity, which, though of a rotational character, shall yet make the whole 
vis viva for the inside and outside together less than that previously obtained. At the 
same time this vis viva is by Thomson’s law necessarily greater than the one we seek. 
Motion in a finite cylindrical tube , the axial velocity at the plane ends (#=0 and x=l) being 
u=u 0 +x(r), (35) 
where 
( ry ^ r yj r= o, (35) 
Jo 
r being the transverse coordinate , and the radius of the cylinder being put cgual to 1. 
If m, v be the component velocities, the continuity equation is 
whence 
du , 1 d{rv) A 
TxM-~dM~ ’ 
dp 
ru =M 
dp 
rv=—fi, 
ax J 
where ^ is arbitrary so far as (37) is concerned. 
Take 
r 
so that 
(37) 
(37'} 
$=u 0 o \ ryfr)dr , 
l . 
u=u,+p{x)x{r), 
l 
It is clear from (38) that if 
v=—f(x)-\ r-yrdr. ( 
Jo J 
(38) 
P(0)=<P(Z) = 1, (39) 
«*= o=Uo+X r=u *= l 
v r=1 = 0 for all values of x. 
Thus (38) satisfies the boundary conditions including (35), and <p is still arbitrary, 
except in so far as it is limited by (39). 
In order to obtain an expression for the vis viva , we must integrate ii 2 -\-v 2 over the 
volume of the cylinder. 
Twice vis viva=ulvl-\-2uA p(x)dx j yfrjl'zrdr 
Jo Jo 
+ \ pxfdx \ 2 vx(rfrdr 
Jo Jo 
+ i ( q^( f rx{r)dr\. { 
l 
«. 0 
• (40) 
