88 
PROFESSOR W. M. HECKS OK VORTEX MOTIOX. 
Ol’ 
'.'K 
? ~ 17 = 
ifwx---^yy- ‘‘ v- 
\ T 
where y = Xx^ja. 
S 2 oircds on the hounding surface, or interface hetiveen tivo shells. This is the case 
where the transformation (Eq. 26) fails. Taking first the stream spirals 
, \ \Ir (Is 
dr)= - Ti' 
a p a-\fr 
(In 
On a spherical boundary this is zero, except for the X., aggregates, in which, however, 
there is no flow at all, 1’he other part of the sti’eam surface is the portion up the 
polar axis. Here ds = dr and dn = rcW. Therefore 
Twist on axis alone = 
2\ 
■v/rr/?’ 
= - [dr = X. 
Cl Jo 
There is no twist on the spherical boundary. Hence 
Angular pitch of stream spiral = X. 
Next for the vortex spirals. Here there are two portions as in the former case— 
the polar axis, and the spherical boundary. 
J sin- 6 
C?9 = 
A, 
a ' ~l 
ds. 
Hence, supposing at present we are dealing with a singlet only 
2X r“, -J , . 2x J\.ad9 
2X r“, -J , , 2X 
? = — i V-VT/ • + — \ 
a J 0 ^ ,1 _ .x-J' a \ 
« shi 0 (J - 
Jo^ — ^^0 / T ^ 
2XJX dO 
0 sin 6 
<dx 
AJ - 
Both these integrals become infinite at the poles. XX^e must therefore treat this 
part separa,tely. 
, p-f J • , 2XJ(X) d- de „ X pfi'- JsiiH^ds 
^ J 0 J - 3/J' X sin X - 3J (X) J„ sin ^ ^ n I 1 (d^f (d<r'\-\ % 
Jfia' ^ U dr ) \rde) J 
