404 
PROFESSOR M. J. M. HILL OH THE MOTION OF FLUID, PART 
The value of-+V is 
P 
2 ab{r 2 - 0 L z f- 2 a\z-Zy+Saba 2 (z-Z) z -(z-Z)Z+e'(t) 
where 0'(t) is an arbitrary function of t. 
The differential equation in xjj has two independent integrals one of which is 
and the other 
ar 2 (z— Z) 3 +6(r 3 —a 2 ) 3 = const.=e 
dr 
t- 
•f 
9 /~ / --sTH— Const. — f 
ciy/ e — b(r' i —ct.~y J 
where after the integration is performed, the value of e must be substituted for it. 
The differential equation in y when solved will give 
y= ar z (z-Z)- § ci(z — Z) 3 + 46a 2 (z — Z) + zZ — J JrZ ' 2 dt-\- <f>(t ) 
4 V + ctbl 
■ 26(46 -\-a)ct.H- 
-f 
r*dr 
y/ a J y/e-b(r*—a?) 
where G and <[> are the symbols of arbitrary functions. 
Finally, in order to express r as and w as G(e, f) is taken as 
(8 6+2a) (6a' 1 ' — e 0 )yi 
Then \= (86 + 2a)(e 0 — e), xp= f, y= above expression with the value of G(e,f) taken 
as (86 + 2a)(6a 4 "— e 0 )f; where e= ar z (z —Z) 2 +6 (r 3 —a 2 ) 3 and 
f =t ~\ 
dr 
y is therefore 
ar^(z— Z) — §a(z—Z) 3 +46a 3 (z—Z)+zZ + 
J 2 *J~<Xy/ C —&(?' 2 — a 2 ) 2 ’ 
46 2 + 
j - 7- 4 rfr 
J a/ c — b(r~ a 2 ) 2 
y/ a J y/e — b(r~ — x~) 
—£t \ f dv 
— (46+a) - —, ,, „ ,+ an arbitrary function of t. 
\ V co 1 J v e — o(r" — ci~y ° 
The value of ——+,—'+- ^ 7 -+ ~ ) calculated from this is complicated. The writer 
47r\ar 2 r dr dz~ J i 
has not succeeded in applying any of the methods of this paper to complete this 
example. To complete the solution it would be necessary to find a value of the 
velocity potential </> which is continuous with y all over the surface e=e 0 , and then to 
examine whether the rate of variation of <f> is equal to the rate of variation of y 
normal to surface e=e Q . The former part of the work is always theoretically possible, 
but it may happen that the latter is not. 
