of a long narrow cylinder 387 



the axis may be taken for that at the surface in approximations of the kind 

 here made. 



We have next to find the integral of the product of the density into the 

 potential. We may consider the product of each pair of terms by itself. If we 

 write ^i for the value of L when = /, or approximately 



(31) 



,3,, 



- - iA t A t l, 



The charge is E = $Xdx = zAJ. (33) 



Determining A 2 so as to make /(A,, + A-J (fa + fa) dx a minimum, we find 



A x A n -* 



^2 ' T"0 T 101 ' 



* 3tr 



and we obtain a second approximation to /f , 



I 



(34) 



36 t< - W- 



This approximation is evidently of little use unless the length of the cylinder 

 considerably exceeds 7-245 times its diameter, for this ratio makes the second 

 term of trie denominator infinite. It shows, however, that when the ratio of 

 the length to the diameter is very great, the true capacity approximates to the 

 value of K given in (18). 



We may proceed in the same way to determine A 2 and A so that 



/(A,, + A 2 + A 4 ) (^T O + ift^ + ifi t ) dx 



shall be a minimum, and we thus find a third approximation to the value of 

 the capacity, in which 



u _ : '3 73 Q _ ill 



^4 yj v -rsis~ ^4 _ ^9 ^ 



101\ lu _ ^9\ _ 45 ' "4 ' ^TT^'O /g __ 1 u II \ /y 889\ 45 



so that when \i is very large the distribution approximates to 



The value of the inferior limit of the capacity, as given by this approxi- 

 mation, is 

 K , 



5 



~ 



__ __ _ _ 



~,f. <. 101 ^ru-i (.1 101\I7tt 101\/o 98\ 461 



3 * ~ -^TT 4 U - TnrJ LI* TnrJ I* ntm TWJ 



As increases, /^ approaches to the value found by the first approximation. 



252 



