382 Note 1 2 : electric capacity 



We thus find for the values of these coefficients 



2 2 2 f" ^ 7 23^ ~I 



A.= - B - - * 3 i - ^^ x'' - 2v* + i* 4 + ^ * 2 y 2 + 3v 4 - &c. 



v 3 L 7 7 J 



? - v 3 [i - 2* 2 - i^y + 3* 4 + 3 - 5 



Lv 4 -&c.] 



We are now able to express the energy in the form 



where A and B are the charges, and p u , p 12 , and p Z2 are the coefficients of 

 potential, the values of which we now find to be 



77 i 2 s 6 T a 2 12 6 2 a 4 a 2 6 2 2 . 3 . 13 b* 



Pu= - -a 1 1 4 3 - --J+IO-J-+I2--+- ^ -j - &c. . 



2 TT 3 2 . 5 c 6 L c 2 7 c 2 c 4 c 4 5 . 7 c 4 



(a 2 + i 2 ) 

 T~ 



i i a 2 +6 2 i a* + b* 2 2 6 2 I fl 6 + fc 6 

 i ~ 3 3- + 5 jr- + 5 -p- - 7 C 7 



12 a 2 6 2 2. 3. 13 a 4 



~ 



NOTE 12, ART. 151. 

 Ow <Ae electrical capacity of a long narrow cylinder. 



The problem of the distribution of electricity on a finite cylinder is still, 

 so far as I know, in the state in which it was left by Cavendish. It is sometimes 

 assumed that the electric properties of a long narrow cylinder may be repre- 

 sented, to a sufficient degree of approximation, by those of the ellipsoid inscribed 

 in the cylinder. The electrical capacity of the cylinder must be greater than 

 that of the ellipsoid, because the electric capacity of any figure is greater than 

 that of any part of that figure. 



It is easier to state the conditions of the problem than to obtain an exact 

 solution. 



Let zl be the length of the cylinder, and let b be its radius. 



Let the axis of the cylinder be the axis of x, and let the origin be taken at 

 the middle point of the axis. Let y be the distance of any point from the axis. 



Let Xdx be the quantity of electricity on that part of the curved surface of 

 the cylinder for which x is between x and x + dx ; we may call A the linear 

 density of the electricity on the cylinder. 



