MR. ¥. W DYSON ON THE POTENTIAL OF AN ANCHOR RING. 
81 
Again, the part of the velocity of the fluid arising from the term A^J^Tir in the 
stream function along an arc of a very great circle whose centi'e is at the origin, is the 
limit of- 
Aj P sin 6 (c“ — cr sin 6 cos cf)) 
= limit of- 7~ 5 —r—A—^, cos (f) d(f) 
r sin ^ J 0 (r^ He- — Icr sin 6 cos (/>)« ^ ^ 
TT AjC sin 6 
2 r® 
Therefore, the part of the circulation rising from the term of the stream 
function is 
ttA^c , , G 7rA,csin0 
TT 
= " A,. 
d 
Now the part arising from or — AjCr— is obtained by differentia- 
c tic 
tion with respect to c, and 
_ A o SVT 
— "^ 2 ^^ g3 ' 
Therefore, the circulation is given by 
27r I 
K ~ 
IA fl- AgC^ -j- 1 . 3 A^cr'^ -h 1.3.5. A^ c*’ fl- . . . |, 
agreeing with the result already obtained. 
We also see that the velocity of the fluid at the centre of the ring is 
'TtA. , „ d ITT 
c dc \c^ 
-f- . . . — {Aj -h 2cr'Ao + 8 (t‘’'A 3 -f- • . . 
The kinetic energy of the cyclic motion may be obtained simply in the following 
way— 
fZfl) 
2T = , If A 
the integral being taken over a barrier. 
This 
1 d-^r 
dvj 
dzs d (f) 
= 'IirpK 
dzs 
MDCCCXCIII.—A. 
= ‘llTpK (l/lR — 
M 
