PROFESSOR W. M. HICKS OK VORTEX MOTION. 
81 
At the equatorial axis Jc =■ 0, on the surface k=]. Thus k increases from 
0 to 1 for the various current sheets in order from the axis to the surface. The pitch 
ol the helix on any sheet is 
Pitch = X \/l — E. 
At the surface this is X, at the axis it is 
= X\/l — 
?/o 
TT 
ttX / 
2 'V 
l-i(l 
112/1 2v/2\ ^112/' 
Since ^T/{'2^y2) =■ ITl, the pitch at the axis is about 11 per cent, larger than on 
tlie surface when X is small. 
The corresponding quantity for the vortex filaments is given by 
= 1 + iJ'J' f 
J .ri 
clx 
By what has immediately gone before 
'5 — 7} = 
ixj' r 
■ ^0 
xdx 
Xi 
, 1 ' 
15. _ i - 
I' z'® 
i_5 A r _ 
1 - ^ (P- + 1) I- v/{(l - X^) ix? - {xi - .fib} 
cly 
- ^ ) V { 0 -- y) iy - Vi) ( 2/2 - y)) 
(10 
28 28 ^ ^ 
15J^ 
2xvr^ 
and 
^9 
IF 
(1(f) 
^ (1 + Vi) - ^ (y-i - yi) shfi I' \/ (1 - F shfi d)) 
28 
J' = ^XM1 -tAF. 
Therefore 
9 — 17 = 
5 1- 
A 
10 
where 
Thus 
2X \/ 1 - 2/1 { 1 - ^ (1 + 3/1) 
f_ 
Jo(l - 
(1(f) 
n sin^ </)) v/ (1 — P sin^ </>) ’ 
^ - 2/0 = 4 F(i - 44 
28 
X /- n,/7 ,, ^ 
"7 7 ^^ 2x71-2/1 
n (— n, A (f)) 
(38). 
At the equatorial axis n = 0,k — Q 11 = 77/2 for a half turn. 
VOL. CXCII.—A. M 
