PROFESSOR W. M. HlCitS ON VORTEX MOTION. 
63 
J' = 
sin A, . 
--COS A, 
X 
cU' . sin> \ \ \ 
~ = A sm A-P cos \ \ sni A 
ax A 
the two ecjuatioiis for Ao, B 2 give 
A., =1 
3 V , 
W—:- 
siu A 
YA / 3J' 
A- \A siu A 
The aggregate moves through the fluid with 
a velocity of translation given by 
u 
2B, 
2ito? 
Y_ / 3J^ 
ttA- \A sin A 
By its formation the above value of \jj satisfies all the equations of condition except 
that in those equations ifj is the velocity-function referred to fixed axes. Here it is 
not—it represents the motion referred to the instantaneous position of the sphere. 
It is, therefore, not directly applicable unless the velocity of translation given by it 
vanishes, that is, unless 
J' — ^A sin A = 0. 
If A be a root of this equation we get a steady motion of a vortex aggregate, at 
rest in the surrounding fluid. 
If we, however, take the above general function, it gives a velocity of translation 
U = 
A" 
ttA' 
3J^ 
A sin A 
(17). 
Bring the aggregate to rest by impressing a velocity—U on the whole fluid—that 
is, add to the stream-function a term — 7rUp“ = — TrUa'^x^ sin 
We get a new value of ip, referred to axes remaining fixed, viz.. 
xp = 
3Vft^ 
A^ sin A 
(J - x^J') sin 
Take this value of ip, and put f= -xp. Then equations (L, 3, 4, 7) become 
cc 
Vp cos <f) 
vp sin (p 
ojp CO.S y 
o)p sin X 
I dxp' 
27 r d n 
A 
27r« 
^ cljr 
47 r« dn 
3 V 
TttA sin A 
J sin’^61. 
