of a long narrow cylinder 



385 



In the case of the cylinder, if we suppose A to be uniform, and neglect the 

 electrification of the flat ends, 



.4 



E = 



W = 



(log f - 



(17) 

 .(18) 



When the length of the cylinder is more than 100 times the diameter this 

 value of the capacity is sufficiently exact for all practical purposes. The capacity 

 of the inscribed ellipsoid is , 



To obtain a closer approximation let us suppose that the linear density A 

 is expressed in the form A,, + A x + &c. + A,-, the general term being 



A,,^ n P,,g), (19) 



where P n is the zonal harmonic of order n. 



If we consider a line of length 2,1 on which there is a distribution of electricity 

 according to this law, and if /j and / 2 are the distances of a given point from 

 the ends of the line, and if we write 



q - t 

 r 



1/2 "/I 



:>) 



2 I ~2 I 



then the potential, ift n , at the given point (a, |8), due to the distribution A n , is 



where P n is the same zonal harmonic as in equation (19), and Q n is the corre- 

 sponding zonal harmonic of the second kind*, and is of the form 



I T 



n (a), (22) 



a - I 



where R n (a) is a rational function of a of n i degrees, and is such that Q n (a) 

 vanishes when is infinite. The values of the first five harmonics of the second 

 kind are 



<?o () = log 



+ I 



<?, (a) - a log " - 2, 



(23) 



Q y (a) - (fa 3 - fa) log * - 5 2 + |. 



Q 4 (a) - (V- - V 2 + I) log "4l - <* + *! 



In applying these results to the determination of the potential at any point 

 of the axis of the cylinder we must remember that a point on the axis is at the 



* See Ferrers' Spherical Harmonics, chap. v. 

 c. P. i. 25 



