386 
PROFESSOR M. J. M. HILL ON THE MOTION OF FLUID, PART 
Adding to this the cyclic term 
cab x 
T SU1 </x* + (y-Yf 
giving a cyclic constant — rrabc=nab (-j- 2 £) where 2£= — — the expression 
obtained is 
c a —b . _ T . , cab . 
2a + i x (y~Y+Y sm 
V x 2 
a* 
+ 
(y-Y ) 3 
b 2 
This will not agree with the value of y, unless 
c a—b _a 2 — b" 
2a + b~ a % 2 ^ 
cab _/(<x 2 + & 2 ) 
Y = aft 
Y=0 
The first and second equation require «=&. 
Hence this method will not lead to a determination of the irrotational motion out¬ 
side the cylinder. It does not prove that there is no irrotational motion outside 
continuous with the rotational motion inside the cylinder. 
Supposing Helmholtz’s method applied to this case, it would be necessary to find 
Q 
a value of A which is the potential of a distribution of matter of density — — inside 
the surface 
^ Ay- Y) 2 
a 2 ' b 2 ~ ' 
The result is that inside, 
A = 
cab 
and outside. 
C-f 
a(a + b ) b(a + b) _ 
cal 
a= t 
" C '+ log <✓*•+«+ 
where 
** (y-Yfi 
a~ + e' & 2 + e 
= 1, and C and C' are constants. 
If the constants C and O' be properly determined, these expressions will be 
continuous at the surface — + ———=1. Their differential coefficients are also 
continuous. 
