ELECTRICAL CAPACITY OF A LONG NARROW CYLINDER, &C. 673 



where / and / 3 are the distances of the point (f ) on the axis from the positive 

 and negative edges of the curved surface. 



At the middle of the axis, 



=0, and /,=/,= 



1^0) = 2X log " + 47TO- (/- I). 



At either end of the axis, = Z, / = &, / 2 = 2/, nearly, 



i/ w = X log y + 2-rra-b. 



Just within the cylinder, when is just less than I, 

 d\jt /I l\ 



= ~ ~ + 27nr = by 



Hence 2ir&cr = X, 



or the density on the end must be equal to that on the curved surface. 

 The whole charge is therefore E=2rrba- (2l + b). 



The greatest potential is i/ {0) = 2Tr6cr ( 2 log -,- + ^). 



V b I] 



The smallest potential is that at the curved edge, and is approximately 



and the capacity must lie between 



E 2l + b E 



i~ = Xi r an d r~ = ri ^ 



y/(0) i 21 U Y(e) 4e 2 



3 b 7 b TT 



These are the limits between which Cavendish shews that the capacity 

 must lie. When the cylinder is very narrow, the upper limit is nearly double 

 the lower, so that we cannot obtain in this way any approximation to the 

 true value. 



To obtain an approximation, we may make use of the following method, 



in which we neglect the efiect of the flat ends, and consider the cylinder as 

 a hollow tube : 



VOL. II. 85 



