PROFESSOR M. J. M. HILL ON A SPHERICAL VORTEX. 
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These are equivalent, on elimination of — + V, to 
P 
~J_I dr 
dwV 
_r\dz 
= 0 
and 
(rr) + ^ (riv) = 0 
(VI.), 
(VIL). 
Art. 2. The Equation satisfied by the Current Function. 
From equation (VIL) it follows that a function x\s exists, such that 
rr = cxjjjcz 
ruj — — cxp/cr 
Substituting in (VI.), it follows that 
/ 0 1 0-v/r 0 1 Syjr 0 \ 
I dt r dz 0?’ r dr dz j 
A & 0V _ 1 dfi 
\ dz~ 0r- r dr 
= 0 
(VIII.). 
. (IX.). 
Hence, the whole motion depends on the current function ijj defined by (IX.). 
Art. 3. The Particular Integral selected. 
The following is a particular integral of (IX.) :— 
d~yjr 0"-v^ 1 d\fr /8k 2A,’\ 
dz^ ^ ~~ T'dh-^ ^ VI • • 
where a, c, k are constants. 
A particular integral of (X) is 
^ = e {|r (r— a'^) + C - Zf +/(()| . 
where Z and fi{t) are functions of t only. 
Substituting this value of i// in (VIII.), 
7= a|-r( 2 - Z), 
w = - 2 J {z-Zf-2~ (2r' - a-) - 2f(t). 
2 F 2 
. . (X,), 
. (XL), 
