480 | MR. 8S. S. HOUGH ON THE OSCILLATIONS OF A 
We have, in addition, the equation of continuity 
Ot 0x 4-500) 04 OW) 02z-— 10) ay. hae uy Eee) 
Equations (2), (3) are sufficient to determine u, v, w, p, subject to certain boundary 
conditions. 
From (2) we obtain 


we oh, eee ov 
|e ee | Pi rr an Oe 
e el, ¢¥ Ow 
ap +- 10°| v erah a 2 A b (4). 
ES ae | 
ot? ~ oveé J) 


> OW 0 
= EOF ae = (Vth); where = tite 
or, by the third of equations (2) 
9 Opp oe D) 
de a + ap (Vb) = 0 st 2 5 Selah TERRE aes 
This is Porncar®’s differential equation for the oscillations of a mass of fluid about 
a steady motion of pure rotation. 
Let us now suppose that the system is executing one of its component harmonic 
vibrations. 
Assume that 
ens v= v,e™, W i= Wier 
y= We. 
and 
Putting these values in (4), (5), and dividing out by the time factor, we get 
eotece, - OWT ov 
i PER. {id oe + 20 ~ | 


oy 
Lf ove oe 
= N= — 20 —= eS oll ee 6 
Ui pepe {in ay on r (6), 
1 ah | 
WW), = = ae 
% in O82 J 
while y, satisfies the equation 
tn 4 Oy do?) Oy 
met ga +(1- ) ao). 2 a 



