1082 
MR. F. W. DYSON ON THE POTENTIAL OP AN ANCHOR RING. 
T / 
^ CT 
Ttt 
(c — xf cos dx dz d(j) 
jVI 
47r 
?T7 
.y/ (cr^^ — 2vy' (c — x) COS (f) + (c — xy + (z' — zy} 
cos (f) d<f) 
^ x{dldc) z{djdz) 
{m'" — 2ct^' cos <^ + + z'~'\ 
( 60 ). 
where the double integral is to be taken all over the area of the cross-section 
E = a (1 + cos 2x + ^3 cos 3x + ^4 cos 4x). 
§31. While finding this double integral we may treat djdc and d/dz' as constants. 
Put 
Then 
The double integral 
taken over the area of 
— = V cos a = V sin a. 
dc dz 
d^ d^ 
dc^ dz'^-' 
{j:c;osa+isin o) 
dx dz 
E = rt (1 + ^2 cos 2 x + ^3 cos ox + cos 4x) 
may be conveniently calculated in three parts. 
( 1 ) Over the circle of radius a ; 
( 2 ) Eound the circumference of this circle to obtain the first powers of /Sj, /3^; 
( 3 ) To find separately the coefficient of 
Part (1) 
where 
where 
(.';cosa + : sin 
dx dz, 
0 I 0^0 
a;-' + z-' < cr 
(61), 
f- + C- < 1, 
= Tra" h e v/1 “ C‘‘Zi 
= 4<r 
J f 
e cos^ 0 c/0 
_ sill I 
— wa p + ^ + 25;|s „ + • ■ ■ 
(62). 
