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PROFESSOR M. J. M. HILL ON THE MOTION OF FLUID, PART 
is part of one of the surfaces \= const., and therefore always contains the same 
particles. 
x'~ ?/ 2 
Hence, if the smooth hollow rigid cylinder —— ; —— =1 rotate about its axis with 
uniform angular velocity d>, such that 
/ « 2 + & 2 0 
\ ct 2 b 2 / 
_4_ 5 V s 
f) a 2 b 2 
x ' 2 y n - 
it is possible that the fluid inside the geometrical surface —= 1 should move 
rotationally, and that the fluid between the two cylinders should be moving irrota- 
tionally, the rotational and irrotational motion being continuous. 
The components of the velocity of the rotational motion parallel to the axes of the 
sections of the cylinder by the plane of x, y' are 
and 
The components of the velocity of the irrotational motion in the same directions are 
dA d\ 
dy' dx' 
i.e., 
and 
aby' 
9 7 0 
ar—o* 
+ e 
6 2 + e' 
ctbx' 
a 2 — b~ 
[ab ab J 
J4 f . a 2 + 5 2 ] 
\ab W ab J 
These expressions for the velocity will not agree with those obtained by Helmholtz’s 
Method, viz., Vvt! - 1 ) and ,,= lS( VSt;- 1 )’ ™ leSS * = -*’ 
i.e., in the case of Example IT., and the irrotational motion as obtained by Helmholtz's 
Method is not continuous with the rotational motion. The other method suffg’ested 
in this paper (in which the potential of the density — 
±(*LX+ d LX 
dtf 
is calculated) does 
47 t \dx 2 
not lead to a result. 
13. Example IV. To consider the case in which the vortex sheets are coaxial 
circular cylinders, and the molecular rotation is a function of the distance from the 
axis. 
This investigation will illustrate the reduction of the components of the velocity to 
Clebsch’s forms. 
