64 PROFP]SSOR W. M. HICKS OK VORTEX MOTIOX. 
These give v, w, <^, ;>(. 
Now substitute in voi sin + X- result is that 
o'^r 
V(D sin (d) 4- y) dn = ^ ^- J'.ddj 
^ Sirki-X sm X ^ 
SO that the motion given by the new ijj is a steady one. There exist, therefore, 
systems travelling through the fluid with velocities given by (17) and with a steady 
motion. The system given by J' = sin X is contained as a special case, 
17, There are two circulations to be considered. That along a circuit up the polar 
axis and down over the surface of the sphere, and that due to the motion I'ound the 
polar axis. Call them respectively the primary and secondary cyclic constants, and 
denote them by p,, p. 
= d“{ ^ *+2 n - 
Jo [^Trp rdO J 0=0 Jo L ^Trp dr J ,.=„ 
In finding this the term siid 6 may be omitted as giving no circulation, and 
we may take 
, 3Vfd -r . 2^ 
ijj — —r J sm u 
p = 
X® sill X 
3Vrd 
ttX® sin X 
8V« 
ttX- sin X 
g'J , dJ' f 
J 7 dJH 
2 — — — I 
hr dy\ 
r‘' a m 
sm u du 
dr J 0 
where 
Now 
therefore. 
y = 
\r 
fg _ i + fl y = _ 1 + f(SBl _ 4) ^ 
J r ^ y i y dy ^ y h y rJ'^ 
^ c J 7 r JI c sfjiy 7 
2 — cZy = — — + —~ dy. 
hr ^ L 3/Jo Jo 2/ 
Also, J (y) is of the order if when y is small, therefore, 
0^ J , 4' 
and 
J 
2 -dy = 
hr 
-j” StX, 
3Va 
p = 
ttX' sin X 
(SfX — sin X). 
If we replace V as a constant of the motion by p, 
irpia 
1/7 = 
X (StX — sin X) 
(J - x~3') sin'^ e. 
