72 
PROFESSOR W. M. HICKS ON VORTEX MOTION. 
lines follow after in the same way, crossing the meridian twice; once in each 
direction. 
Beyond Xi"* we get triplets. 
In general, between Xf^ and the blue lines lie to the right of the red or the 
opposite according as n is even or odd. They coincide for the Xj parameter. Also, if 
n is even, both lie to the right of the meridian for the inner nucleus, the reds to the 
left for the second layer, to the right for the third, and so on. Whilst the opposite 
takes place if n is odd. 
The forms of the spirals may be obtained by finding the polar equations to their 
projections on the equatorial plane. Let (p, rj) be the polar co-ordinates of a point 
on the projection of a flow ; ( p , 9 ) of a vortex line lying on a given sheet x}j. Then 
(^■v . j d? • 
where ds is an element of a meridian curve. Hence 
d-yfr 
P dn 
\ dr 
a d^ 
P VW 
Provided dr is not perpendicular to ds, i.e., on the outer boundary, but then xjj — 0 
and 7) = 0. 
dr 
2 cos 6 ’ 
’X, 
dx 
-’ > 
cos 0 
where corresponds to the inner circle of the two in which the current sheet xp cuts 
the equatorial plane. The total angular pitch of the spiral is 
X 
I 
dx 
cos 0 
where x^, Xo are the two roots of 
T/s \ oTx X\/r(Si:x — sin \) , 
J (X.r;) — aj-J X = - = b, say. 
iTjjLa 
The above may also be written 
( 26 ). 
(27). 
Equation (26) enables us easily to determine the form graphically when the 
surfaces \p are drawn. So 
5 = 7? + 4 J' r ^(J - J') (J - J' - dx . . . ( 28 ), 
Jari 
the case of a spherical boundary being excepted as before. 
