70 
MR. F. W. DYSON ON THE POTENTIAL OF AN ANCHOR RING. 
{d) d^jdn has a definite value at the surface of the ring ; 
has a definite value at the surface of the ring. 
The different cases must be considered separately. 
Let the figure represent a section of the anchor ring through the axis Oz, at an 
azimuth (ji. 
z 
Let P be a point on the surface and let ^ OOP = 
(i.) When the ring moves parallel to Oz, with velocity w ; 
— = tysmy. 
an ^ 
(ii.) When it moves parallel to Ox with velocity u, 
~ — u cos y cos <&. 
cm ^ 
(iii.) When it rotates round Oy with angular velocity cu^, 
fin 
= — Cwo sin y cos (f). 
The stream function for motion parallel to Oz, with velocity iv, satisfies at the 
surface the equation 
(i.) = 0-\-^ ID {c — a cos x)’^- 
For the cyclic motion, 
(iv.) = C' at the surface of the ring. 
The formulae of Section I. show that we may take 
«!> = |Ai + Aocr+ &c.| in case (i.) 
<t> = [A^Jj + A^n^Ja + A^a'^Jg- + &c.} cos </> in case (ii.) 
r . . j .n 1 I' \ 
q> = 1 Ai - + A.xd —r + AoU^' + &c. 1 cos 6 in case (iii.), 
^^dz -^dz -^dz '' ' 
