750 
PROFESSOR W. M. HICKS ON THE THEORY OF VORTEX RINGS. 
47 TajXcfV'V — ^ix.^d' —~ + 2Ln — 1 
■£(/*!+tfd { 3 (^ll — 32 —+ x ) — i/(l + x ) — (Vl — + )| 
• (36) 
^ = I ^ +/)*<* } (1 -x) + tfd(± -1 
+ 2 / /j//+(/x 1 — Y^y) lo sl ~\d .(37) 
Discussion of results. 
9. (/3 X ). The expression for /3 ] gives the relative position of the internal hollow to 
the outer boundary of the core. The centre of the cross-section of the outer boundary 
is outside that of the inner at a distance (by 5) 
= 2a(y-V+A^)=2aV(l-*+f 1 ) 
This is, as we shall see, in general negative. Hence the centre of the inner lies 
outside that of the outer, at a distance whose ratio (y) to the radius (2 ak 2 ) of the 
outer is 
y=~ x+ h> 
Substituting for /3 x /& 3 
9 _ 7 
2 Qt*' 
=**M 1_a:+ log log *)} 
=^i+/*Y (say).(38) 
1+3(1— x) 1 
where y x , y' are the values of y when y l is very large, and zero respectively. If 
therefore we know the points at which the centres lie when the added circulation is 
very large, and when it is zero, the actual position is the centre of gravity of t^vo 
masses proportional to the circulations, placed at the corresponding centres. It is 
only necessary therefore to discuss the values of y x , y separately. 
When there is no added circulation 
, 1 /5 — ox x , 1\ 7 
When there is no hollow, x=0, y' =-§& 3 . This is the point at which, if the pressure 
be diminished, or the ring 1 be increased large enough, the hollow will begin to form. 
’ o o o 7 o 
