164 
MR. W. M. HICKS OK THE STEADY MOTION AND 
translatory motion, a result obtained by Sir W. Thomson'" from general reasoning. 
The terms obtained by the second approximation arise from the translatory motion. 
In Art. 13 the time of vibration of the steady form is obtained, when the cross 
section is crimped, or the whole hollow surface fluted. For this mode the time 
of vibration is, for small rings, given very approximately by pd/(2p^/n), cl being the 
density, and p the pressure of the fluid at a great distance, whilst n is the number of 
crimpings in a section. This, it is to be noted, is independent of the energy, and 
depends only on constants of the ring, and the fluid, and the mode of vibration. If 
the hollow pulsates, or changes its volume periodically, the time of pulsation is 
{pd/2p>)y (log ijk). As k depends on the size of the ring, and therefore on the energy, 
this time is not independent of the latter, but it varies extremely slowly with it. 
The times here given must be understood to apply to the steady motion; when the 
ring is changing its size they must be modified. The investigation of this case, and 
of that in which there is a core of denser matter than the surrounding fluid, I hope 
shortly to take up. 
Section I .—The functions. 
1. The functions whose properties were investigated in my paper on Toroidal 
Functions are only suitable for expressing fluid motions about circular tores when 
there is no cyclic motion through the aperture. It will be necessary therefore to 
investigate some method by which this can be taken into consideration. If we 
consider only motions symmetrical about an axis, and in planes through that axis, it 
is well known that the motion can be represented by Stores’ stream function. This 
function is only multiple valued when there are sources or sinks in the fluid, the cyclic 
constants in this case being the normal flows outwards through surfaces completely 
enclosing the various sources or sinks. If i fj denote the stream function, the velocities 
at any point are given by 
the equation 
— 1 and, when the motion is irrotational, xb satisfies 
P oz p op 
b~-p b~p 1 A 
bf + Tf~~ P ~bp = ( 
To transform this to the independent variables (u.v), where u-\-vi=f(p-\-zi) , we 
notice that the kinetic energy of fluid motion within any space, with given normal 
motions over this surface, is a minimum when the motion is irrotational, or the above 
differential equation is satisfied. The condition is therefore found by making 
a minimum. Now 
* “ Yortex Atoms,” Proc. Roy. Soc. Edin., vi., and Pbil. Mag. (4), 34. 
