PROFESSOR W. M. HICKS ON VORTEX MOTION. 
87 
and 
Write J - x^J' = f 
Let Xo be the value of x at the axis, so that^' = 0 when x — Xq. 
For points near the axis, 
cr = a?o 4- 
f¥) =/+ «/'-f + = /+ iaT'.P, 
<5 — 7) 
” " J-« 
(if") \(^, + r- 
where means the positive value of \ f‘ 
To the first order. 
^=^1= -2 
d^f 
Hence for the total pitch, 
— 7 ] = 
Put 
x.r / 
(Xn + ^ f 
F(/') ^ 
l:- 
[ (ii - P) ( 
! 2 / 
V cdf" 
2 X.P f 
i+fi 
{4 
+ f) 
F (/') 1 
[(S- P)l 
( 2 / 
V F/' 
i * 
sin 6. 
or writing 
cie 
4XJ' 
V(- 2//0 
/y»2 
U/Q 
cc-r 
F + |.E 
At the axis itself F = 0. 
^ ^ = V(- W") ■ 2 
4XJ'a;o 7r 
5=,7\ ^ 
V(- 2/n 
27rX 
V(- vn 
27rXJ (y) 
av/(- 2fn 
2’jT^y 
v/2/(jy - ^ 
{(/) + 4J'} 
if 
if }(y)>ii 
X 
