758 
PROFESSOR W. M. HICKS ON THE THEORY OF YORTEX RINGS. 
Now 
A __ ^l a | /* a r l 
° t r^TT'SZrf-r* 
A a— 
Hence 
47ra 2 (A; 3 2 —A 2 ) 7r(r 2 2 —r : 2 ) 
E,=JBrf{A - j£-/)' log 1+vL- r '^.')+fe»^j +*»<*▼* • • (49) 
Here the terms in // are small compared with the others, even in the case where x 
is nearly unity. Hence the principal part of E : is 
E ! = iR(W log Li-H) 
To the same order E 3 has been found in the paper “ On the Steady Motion of a 
Hollow Vortex ” [ 1 ., § 12]. Using the result there obtained 
where 
E 3 = 2ttV 2 [ U'gi/^o + \ U' 3 V (Bfl—Xa 2 +^r 2 ) + ^RrV 2 ] d' 
pj I__}H_ ^2 
^irak^ 27 rr 2 
4 =^(L,- 2 ) 
Further, since \= 1 — 4 (L 3 — 2)& 3 2 , r z = 2 ah 2 , R=a(l-|-2& 3 3 )' 
R 2 -^ 2 +^=4(L 3 -i)a 2 V=(L 3 -iK 
Hence 
E 3 — 2 77'7 
=M' 
# ‘ ! Y(L i -2) + ^r !! (L a -i)+iEr,V 2 ' 
47 T 2 7 ^ 
/U(l-2Iy) 
4 ttV 
d! 
( L ,-2)+2^(L 3 -i)+iV 3 
( 50 ) 
where M' is the mass of the outer fluid displaced by the ring. The most important 
term in this is 
E,= Mj~(L 2 —2)=i<*Va (4 - 2 ) 
E x is in general of order k 2 with respect to this, unless there is an added inner 
circulation /x^ 
When there is no added internal circulation, the energy of the ring itself is small 
* See erratum at end. 
