MR. V. W. DYSOK ON THE POTENTIAL OF AN ANCHOR RING. 1087 
At the surface of the ring 
E, = a (1 -f cos 2x + &c.). 
Substituting for 5, o-(l + ySgCOS 2^ + . . .) and writing log — — 2 — X, 
(X/ 
xss . . 2\ + 5 , . 12A + 11 
J - = X + 
16 
cr + 
. 2X-1 a:- 
2048“'" “ 4 
+ cr COS X — 
\ + 1 . 3\ + 5 
+ 
A 
■’—7 A 
64 “ 4 
+ COS 2x - + 
Ao-- , 12A + 7 4 p Ap A + 2<,p 
— 0-" - A - 4 o-A +~^^-A 
Acr/Sj'' 
16 
, „ / 3A - 1 „ 
+ cos3x(--^-, o-'-ft- ^ 
, , / 15A-8 , _ A _ 2A-1 
+ cos4x(^- -ft- JO-ft-J,—<7-ft+ -^j+. 
1 <7J If 2A + 3 4A 7 / y . / , 4A +1 ■> , i o 
0 *7 = S ■" - "W + C08 x( - 1 + <-' + i ft 
4 128 
^ a . 4A + 1 
+ COS 2x , + 
64 
0-5 — i o-A H- i /S 3 
+ COS 3x ( ^ + i A ) + cos 4x ( 
1 _^\3 j ct' _ 1 
c dcj '' c cV- 
12A + 0 „ a- 0/1 ^ 
O- — A - 7 cos X + cos 2x 1 — 
32 
2\ 
/ •'> 
/ (7'" 
-cos 3X-COS 4x + A 
oA ^7= '‘X 
'li\4 »' 1 . ^ 
7*] J 7 = 77 . ® 
(72). 
§ 36. Now, in the steady motion, if V be the velocity of the ring. 
Therefore 
is constant when 
^ Vct- = constant at the surface of the ring 
li" 11“ 
'P — ^ V (— 2cK cos X + "7 + ^ cos 2x 
. . (73). 
Therefore 
II = rt (I -j- cos 2x + . . .). 
