82 
PROFESSOR W. M. HICKS OK VORTEX MOTIOX. 
Thus the pitch at the equatorial axis is 
and 
Therefore 
— Ih /I- , 28 
v 1 d~ „ /- - 
^ 2\V1 - 
r- 
7 T, 
2/0 = i ( 1 - 
112 
= .2r/-,{^' + 2{5--AVxT 
= x-Jdi + *n 
112 
If \ = 0, the pitch is oo, as it clearly ought to be, since all the vortex filaments 
then lie along parallels. 
The Form of the S 2 nrals near an Eciuatorial Axis. 
The meridian sections of a current sheet near an axis will evidently in general 
be elliptic. To find rj it is therefore necessary to determine for an ellipse the 
value of 
r dr 
J cos 6' 
The following general theorem enables us easily to do this. Transfer the origin 
to any point O' in the equatorial plane, at a distance c ; and let the new polar 
co-ordinates of a point P be r'.B', corresponding to r.6. Also let x.y denote the 
Cartesian co-ordinates referred to O'. Then 
F — r'^ -1- c“ -b 2cx, 
rdr = r'dr fi- cdx, 
f dr j 
[■ rdr 1 
1 
+ 
r dr' 
1 cos 0 ' 
1 r cos 6 J 
1 r' cos 6' J 
Icos 6' 
y 
For the spirals near the axis the point of interest is to determine the angular pitch. 
Now clearly for a complete ellipse, whose axes are parallel and perpendicular to the 
equatorial plane, and whose centre is at O' 
dr' 
cos 6' 
= 0. 
