PROFESSOR W. M. HICKS ON THE THEORY OF YORTEX RINGS. 
757 
whence 
But 
and 
Now 
2=1-85 
K i'{ z 1 = mil) o 
3 
Kl 12800 x-85 
If _i \ loci' 2 — 1 1 
Y—iZL 2 (s-i ) 3 b 4 (g-i) 
V 1 , 1 +x 
:-594. 
If the pressure had been so great that there was no hollow at all, then 
and 
8 k 3 =P 
i 
400 
V' =J LV-i_ 
V 2 Lo —t 
‘625. 
These cases are however not comparable, their masses being so different. 
L2. Energy .—The energy will be composed of two parts, external and internal. 
Denote them by E 2 , E x respectively, and let 
E=E 1 +E 3 
We need only the lowest order of terms in E x . We can easily, therefore, obtain it 
by supposing the core to be bounded by concentric circles, and the velocity at any 
point to be, V in the direction of translation, and U perpendicular to the radius to 
the centre of the cross-section. The energy will not contain terms in Y U. Hence, 
It being the radius of the axis of the ring, 
E : = l 2 Trite/. f 2 27rrcMP+ ^mc/V 2 
J r x 
= 2^Rd{i^ log Y)+^W-Y)} +>d\> 
5 E 
MDCCCLXXXV. 
