78 
PEOFESSOR W. ]\J. HICKS OK VORTEX jMOTIOK. 
Since 2 is not large (it is of order 1/a), we get 
Write 
a- 
+ W ) ( 1 “ O + IT ) + ( 1 
X 2 
z 
- _L 
6 ' 
(-)■' 
6 ',, 
a. JX 
X ■ X 
1 9^\ 
- + 
a“ a 
(-)-2-^(l + * 
b 
a 
JX 
X 
The greatest value of a/X is < 1. J'X/X is of order 1/X, therefore at least of order 1 V/. 
Hence in the most unfavourable cases h is < 2. The above equation can be written 
1 b hz 2z £- ^ . I, 2^ 
„ „2 „2 0„ « ”t“ 3 b “b “b j,H“ K, > 
a 
cc a“ 
b - 1 
a 
a 
6 
a" 
2; = 
OL 
h 1 
a a 
b-l , + 2)(6- 1) , (b- lb , (h- lb 
5: 
a 
+ 
cc' 
+ 
-i- 
(1st approx.) 
6^3 ^ approx.) 
h — 1 (b — 1) (& + 2)(& 5) 
a 
(36). 
It will be convenient to put h — 1 ziz c. Then 
_ c c(c + 3) (r; + 6) ^ ^ 
a. 6 «'^ ' a? 
c(c + 3)(c + 6)(c- + Jr + 6) 
12 
Sr- _ _ — 
3^ 4 , 
r 
5/ 
If X is a Xi root, c = — 1 and 
__ 7-^ 
a oa’ 15a'^ ’ 
which agrees with the result already found. 
2 4. The Spiral Forms taken hi/ the Lines of Flow and Vortex Filaments. —The 
equations determining these are given in § (21). Unfortunately, however, they are 
not integrable in finite forms. 
We give a graphical method for the stream-lines later. At present it is proposed 
to determine (1) the forms of the stream and vortex lines when X is small, (2) the 
pitch of the spirals near the equatorial axes, and (3) the pitch of the same on the 
outer surface. 
Let the stream surface ip, the streams and filaments on which we have to investi¬ 
gate, cut the equatorial plane in circles given by rja = Xi and .r.,. Then 
dx 
