PEOFESSOE. W. M. HICKS ON VOETEX MOTION. 
73 
For the outside stream-lines the pitch is 
For values of X, however, lying beyond there are several layers in which the 
stream-lines are distinct. If Xi. x^, x^ . . . denote the values of x corresponding’ to 
the interfaces of the layers, the pitches of the stream-lines on those surfaces as we 
pass outwards are 
(xj — 0) X, (xo — Xi) X, (xg — Xo) X, &c. 
\Ye have seen that on these surfaces the stream-lines coincide with the meridian. 
These parts therefore produce no part of the pitch. The twist must be supposed as 
taking place in the part of the stream-line along the polar axes. It is easy to see 
that this is so by considering current sheets near the interfaces. 
We may therefore regard the physical meaning of X to be the criterion of the total 
external pitch of the stream-lines. We will return to the consideration of the pitch, 
and the shape of these lines later. 
The total angular pitch of a stream spiral on any stream sheet if/ can easily be 
expressed in terms of the volume of the fluid inside that sheet. For 
ds XJ' p" TTua 
- . - . ^ - 
p cd d-yfr X (Six — sin X) 
dn 
XJ' 27rp dsd?i _ 7rp.« , 
27rcd d-\Jr' ^ ^ X(SiX — sin X)' 
ch = dy — 
Integrate round the stream surface 
<; = y — 
\J' d f , , 
XJ' dm 
27rcd d."\^' 
(29), 
where m denotes the volume inside if/. 
22 . The discriminating properties of the Xj and X., parameters make it importanc to 
determine their values. The Xj parameters are the roots of the equation 
J(X) = 
sill X 
X 
cos X = 0, or tan X = X 
The large roots are clearly nearly {2n 1)7 
Put 
X = (2?i -\-l)~ — y=a — 
Then 
TT 
VOL. CXCTI.—A. 
COS W 
— sin 1 / = 0 . 
a-y 
L 
V 
