46 
Whence 
PROFESSOR W. M. HICKS OK VORTEX MOTIOX. 
2^ ^ + (/^ A;-) ¥ ^ , 
= 27r I ^k'ah-'- + (^‘ — k') ~ — \k'r^ | siir 6, 
•4:7r {k — k ) 1/’ -1- k'cv' . ^ 
= -Sin- U. 
lo r 
The normal velocity at the outer boundary is 
1 d-v/r , A {k — k') ¥ + //fd 
6- 7^ (when r = a) = ^- 
"lirp rcW ^ 
cos d. 
The outer boundary therefore progresses unchanged with velocity of translation 
y _4, 
— 15 
{k - k') ¥ + //fd 
Bring the outer boundary to rest by impressing on every j^art of the fluid a 
velocity equal and opposite to this, i.e., adding to the stream-functions a term 
- 15 
^ {k — k') ¥ + ¥¥> _2 
CC‘ 
?■“ sin- 0. 
The relative motions are then given by 
i [bk'ct- + 5 (A: - k') 6“ - 2k'a? ~ 2 {k - k') - /;?•-j r- sin- 0 
= ^ -kr--^\{k- k') ^5 - 6^1 n- sin- 0 
xP 2 =j (a- - r-) r- + I (^ - ^') sin- 0. 
If, however, the motion is to be steady, the inner sphere must now be at rest, 
that is rpi = 0 when r —h. We get, therefore, the following necessary relation 
between k, k', a, h, 
k'a^ _ ^ 1 - ( 7 , _ Jc') (^5 - 6 - = 0 . 
This may be written 
2¥k (a® - ¥) + k'{da^ (a^ - ¥) - 2h- (cd - ¥)] = 0. 
Both the expressions in the brackets are positive, hence k/k' must be negative or 
the rotations in opposite directions in the two portions. 
