PROFESSOR W. M. HICKS ON THE THEORY OF VORTEX RINGS. 
759 
compared with the whole. When, however, there is such a circulation, and when the 
hollow is small, the internal energy is large, and may be larger than the external. 
As, however, the aperture increases—or as we have already seen the hollow increases 
—the internal energy will rapidly diminish. The effect therefore of increasing the 
whole energy is to actually decrease that of. the core. 
The external energy will, as in the ordinary theory, increase with the aperture. 
Section III .—Small Vibrations. 
13. The vibrations here considered are (1) when the core is fluted, and (2) when it 
pulsates. The normal modes can be represented by principal displacements of the form 
a n cos nv, together with a series of terms whose amplitudes are (ivith respect to Jc) 
infinitesimally smaller than a n . The time of vibration will be a function of a, k x , k 2 . 
and may therefore be supposed expressed as a series in terms of k lf k 2 . It will be 
the aim in what follows to obtain the first, or principal term, in this expansion, in 
other words,—in functions of k 1} k 2 —, k x k 2 will be regarded as small quantities of the 
first order. To this order it will be found that we need only consider the principal 
term in the normal mode concerned. 
Let the form of the surface at any time be given by 
k=k x {l-\- a n cos nv -f- y n sill nv) 
k— &o( 1 + /3 U cos nv-\- sin nv) 
where a, /3, y, S are functions of the time. 
The motion is then determined by the stream functions already obtained, with an 
added stream function, or velocity potential, to give the small motions. 
In general, the stream function for the vibrations will be many valued. It has 
however been shown (I., § 13) that in the case of fluted vibrations this portion is of 
order k n+1 , and may be neglected. This cannot, however, be done in the case of 
pulsations, and for that the velocity potential must be employed. 
Let Xj, be the additional stream functions. 
Then 
Xl vAC-f 
X2= V(c— 
(A W RR B,/Ih) cos nv -j-(OIL-f D«T«) sin nv] -j e 
A'JR„ cos nv -f C' «R» sin nv }+e 
where e, e' are uniform along the boundary. 
The expression for the pressure at anv point of the core has been already found 
(11) viz. : 
f= c +/(0 “ <R iC vel )R-R- 
5 E 2 
