OF WHICH IS MOVING ROTATION ALLY AND PART IRROTATIONALLY. 407 
To find the potential of the shell, put 
'i_1 
C xy a? — /3 2 « 1 c '\b 2 a 2 
where 
87t a 2 /3 2 8 x 47r Ac 2 
ct—ma , /3=mb, Sa=aSm, —+7;=m 2 
«•" 0 Z 
Hence 
C=-2c- 
8m 
m" 
Therefore potential inside is 
2 c8m , a — b 
■^r*^b x y 
and potential outside is 
2c8vl ah 
w?a? + to 2 Z> 2 + 2e 
m 3 & 2 — « 2 \ L 2y /, (m 2 a 2 + e)(m 2 & 2 + e) 
xy 
where e is given by the equation 
r 
ra 2 a 2 -fe m 2 & 2 + e 
= 1 
To calculate the potential of the whole cylinder, put a—m -P so that 
^ , r 
« 2 + I )_t > + P' 
:m 4 
Then the potential of the shell, considered above, at an external point, becomes 
2 c8m ab / « 2 + fe 2 + 2P 
" b~ — o? \ ”2 v / (« 2 + P)(6 2 -f P)^ 
The limits of integration for m are 0 and 1, when integrating to find the potential 
of the whole cylinder at an external point. 
Therefore the limits of P are 00 and that value of e which satisfies the equation 
a: 2 7/ , 0 0 
and which makes both cfi+e and Z> 3 +e positive. 
Also 
2 m8m— 
r 
(a 2 + P) 2 ' (6 2 + P) s 
SP. 
