PROFESSOR W. M. HICKS ON VORTEX MOTION. 
83 
Further, if the axes are a, respectively in the equatorial plane and perpendicular 
to it, 
X z= a sin y ^ cos 6, 
where 6 is the excentric angle of a point on it. Hence, 
[dx _ C+.ra cos 6 dO OL 
Therefore, 
t] (for half-turn) = 
y /3 cos Q " /3 
7r«c \ 
or. 
1 ., , 7r«c N 
angular pitch = — — . 
To ajiply this, it is necessary to determine the form of the current sheets near the 
axis. 
Let the co-ordinates of the equatorial axis be c, o. 
The equation to a current sheet is 
or. 
/A,?’ N 
) - — 
^ ! 
' d- 
f \ 
[j(V 
— < 
r' 
^ ^) !■ = constant. 
a- 
^ = J (X)/X“, and p are nearly = c. 
Denote J -/r by f, and suppose it expressed in terms of x, y co-ordinates. 
Refer to O'. 
Then x = c + y = 0 -f 7}, where f, y are small. Hence, if f now denote the 
value at O', 
0 + 
(C + + yf. 
df df X 
dx dr r ’ 
. 'V , 9^ 'V , = 'T-.f 
+ aXIy + H 
d'J? 
— constant, 
^ ^ J/ 
dy d r r ’ 
and dfjdr = o, for c is given by this equation. 
Denote dfjdr by dfjdr- by f". Then 
~ f + =/"> si'^'ce a: = r to 1st order. 
~ f' f" = 0, since y = 0 to 1st order, 
dxdy 
IL 
df 
0 . 
M 2 
r 
r 
