PROFESSOR W. M. HECKS ON VORTEX MOTION. 
61 
round the two circles (1) the equator of the sphere, and (2) the equatorial axis, it 
is therefore given by 
v = y(2Ai/;'h 
or 
p —. Aa y/(2.'r — I — 
where x is the root of equation (13). 
On account of the artificial nature of the internal nucleus the further discussion 
of this case is scarcely called for. We pass on, therefore, to the more important 
case—the next simplest one—in which F is uniform, but the second terms varies 
as rjj. 
cl f 
15. Case f —Here also f varies as xfj. 
Write f = where a is a length, which may be taken to be the radius of the 
SttV^ * 
sphere, and X is a pure number. Also write F = where V is a velocity. Then 
the equation in ijj is 
(Pyjr 1 d'-\lr COt $ dxjr p- „ V 
^ d^ W le~ ~~ cd ^ V- 
V 
A particular integral is — — C and the general integral depends on 
, 1 d^lr cot 0 d-\p' , ^ 
In this put if/ = where is the function of 0 already discussed (§ G) and J„ is 
a function of r only. Then 
d'-Jn ___ J n(n - 1) 
d,.2 I ,.2 
JL/y r is therefore a Bessel’s function of order n — T, which can, as is known, be 
expressed in finite form involving circular functions. In what immediately follows, 
the values of J 2 will alone be required. The equation is, writing x for r/a, and 
dropping the subscript 2, 
dx^ 
If J and Y denote the two integrals 
