SMALL VIBRATIONS OF A HOLLOW VORTEX. 
177 
V 
60° 
90° 
120° 
V 
Y> 
•125 
•355 
•652 
180° 
1 
For a ring-shaped surface 
u— 2-9662, v= 180, and Y/Y 0 =1'699 
whilst for a negative translation 
u— 2'9662, v—0, and V/Y 0 = — ’3708 
are sets of corresponding values. 
8. The energy of the fluid motion .—The energy is given by 
supposing the density of the fluid to be unity. Treating this in a similar way to the 
analogous expression in terms of the velocity potential, and remembering that when¬ 
ever the volume of the surfaces immersed remains constant, as here, is single valued, 
we shall find (by means of equation 1) that 
the integration being extended over any meridian curve of the solid, and dn being 
measured inwards along the normal (i.e., from the fluid). 
In the case, therefore, of circular tores 
byjr bu dn 
bu bn dv 
2,r \fr b\lr 
— dv 
o p bu 
Now we know that for the cyclic motion the energy is cyclic constant X flow 
through the aperture, and, therefore, with our notation is /x X 7 t\Jj 0 . But it will be 
interesting to see how this is also arrived at from the preceding expression. The 
whole energy can be put in the form 
E = (a 2 yyx)\fj 0 2 
we proceed to determine a, ft, y by means of the above formula, 
a. Here along the surface i/;=i/; 0 a constant, and, 
MDCCCLXXXIV. 2 A 
