158 
PROFESSOR H. LAMB ON ELLIPSOIDAL CURRENT-SHEETS. 
the corresponding value of (f> will be 
4>= 2C (1 - fi) 
where, by the formulse analogous to (69), 
A = ( — )' 4ir 
= - 2 tt 2 C 
= - 2tt 2 C 
■2\c/9 d If (d') 
a - nr \c «?/■•» ^ 
COS set), ■ 
(96) 
'pi — s 
duf 
n + 5 
jK o 
| n — s 
] 
| n + .s ’ 2 . 
?=0 
• C, 
1 . 3 . . . (n — s) 
2.4 ...(% + s — 1)' 
• (90 
II now, to use Maxwell’s artifice, we pass to axes of x, y, moving with the disk, 
we must write 
^=C(l-^^^COS 8(0,+pt), 
where p is the angular velocity of the rotation. For the trigonometrical term at the 
end we may write e is( “ +1>i) if we retain in the end only the real part. Hence for the 
induced currents we have, by (91), 
— ispr — 
^ 1 + ispr < ^’ 
where r is the persistency of free currents of the type (n, s ). 
Putting 
y = arc tan spr, 
we find, finally, on returning to fixed axes of x, y, 
<t> = C sin T) (1 - d jfi sin (s,„ - ,). . 
. . (98) 
The system of currents is stationary in space, but is displaced relatively to the field 
by a greater or less angle 
- arc tan spr, 
according to the speed of rotation. The maximum value of this is n/2s for a suffi¬ 
ciently rapid rotation. 
* This represents a fictitious distribution of currents which would give at all points of the disk the 
same normal force as the actual field. 
