MR. F. W. DYSON ON THE POTENTIAL OF AN ANCHOR RING. 
71 
and that we may also take 
'ir = r sin 6 + AgCt^Jg + &c.} in cases (i.) and (iv.). 
Aj, Ag, &c., are constants which must be found separately in each case. 
§ 11. Motion Parallel to the Axis of the Ring. 
Let the ring move with velocitji^ lo parallel to its axis. 
The value of the normal velocity is iv sin y at the surface of tlie ring. 
At the surface of the ring E, = a. 
Let cr = ajc =: value of ^ or E/c at the surface of the ring. 
Let X = log 8c/u — 2 = value of I or log 8c/E — 2 at the surface of the ring. 
Let us assume that 
dl. 
-t = A, 4^1 + + he. 
^ dz ^ dz ' ^dz 
Then A-^ Aj, &c., are to be chosen, so that 
i.e. 
d(h 
= tv sm 
dn 
X’ 
d(b 
- = Cic sm X. 
Differentiating the formulae (B) of § 3, we find 
d (dl^ 
ds\dz I cV 
= — i sm X — 
4^ + 1- , g •o\2 f6l + S . ^ . ^\‘:i 
sm X 4- 3^ sm 3x) sm 2x + ah sm 4x) 
31 
^ sin X + ^^^2048^^^ } 
\ 256 
I (§) = "X + (I sin X + i sin W « 
— ( ^^4 ^ X + -AV sin 3x + lia sin 5x) — *>=. | 
d fdlA 2! r . /K . . 1 • \ 
+ (tI sin X + M sin 3x + sin 5x)s® + &c. j 
= 4x + sin 3x + I sin 5x) 5 + • • -j 
ds\dzl 
