PROFESSOR M. HICKS ON 70RTEX MOTION. 
59 
Section iii.— Gi/rostatic Aggregates. 
14. Passing on now to the consideration of the more general problem where a 
secondary spin exists, the simplest case is that in which in equation (12) both F and 
yt/Z/di// are uniform. 
Suppose 
/|" = A, or /= v/(2Ar//). 
The differential equation iii il/ is now 
1 (1-y^ 
dA 
cot 6 d-^ 
dd 
87r"p“F — A, 
a particular integral of which is 
^ - lAr-^ 
and the general integral is the same as that considered in the previous section, viz. : 
( A7‘"' + Z„. 
It will however not be found possible to satisfy the boundary conditions unless the 
term xjj = Ap" be introduced. This term, as well as that in makes the motion 
discontinuous at the polar axis. However, we will suppose for the moment this 
portion of space excluded, and see later if it is possible to do so. The stream- 
functions are then,—inside 
xpi = — tt^p^F — |■A9•■ -f- Ap’ S2 A,p'"Z,,, 
outside 
and can be replaced as before by fr^Z.,. 
Applying the conditions if/i = ij/j and dxjjijdr — dxjjoldr, when r = a it is easy to 
deduce that 
i/;, = - iA {a - rf - iir-VAZo F ^ir-a-YAZ,, 
F. = -A-tt^F ^ Z. 3. 
The velocity normal to the sphei’e is 
1 fZF 
A-TTp rdd 
That is, the sphere progresses bodily with a velocity given by 
= "A’ 7rF(d cos 6. 
