Length of Terminated Rods in Electrical Problems. 5 



admissible values of h, with addition of (1) which in fact 

 corresponds to a zero value of h. And each value of h 

 contributes a part to each of the infinite series of coefficients 

 B, needed to express the solution on the positive side. 



But although an exact solution would involve the whole 

 series of values of h, approximate methods may be founded 

 upon the use of a limited number of them. I have used this 

 principle in calculations relating to the potential from 1870 

 onwards *. A potential V, given over a closed surface, 

 makes 



reckoned over the whole included volume, a minimum. If an 

 expression for V, involving a finite or infinite number of 

 coefficients, is proposed which satisfies the surface condition 

 and is such that it necessarily includes the true form of V, 

 we may approximate to the value of (14), making it a minimum 

 by variation of the coefficients, even though only a limited 

 number be included. Every fresh coefficient that is included 

 renders the approximation closer, and as near an approach 

 as we please to the truth may be arrived at by continuing the 

 process. The true value of (14) is equal by Green's theorem 

 to 



1 M V /Y 

 Itt J J dn 



IV 



rfS (15) 



the integration being over the surface, so that at all stages 

 of the approximation the calculated value of (14) exceeds 

 the true value of (15). In the application to a condenser, 



whose armatures are at potentials and 1 ,( 1 5) represents 

 the capacity, A calculation of capacity founded upon an 

 approximate value of V in (14) is thus always an over- 

 estimate. 



In the present case we may substitute (15) for (14), if we 

 consider the positive and negative sides separately, since it 

 is only at c = that Laplace's equation fails to receive satis- 

 faction. The complete expression for V on the right is given 

 by combination of (2) and (11), and the surface of integra- 

 tion is composed of the cylindrical wall r = h from c = to 

 s=oo, and of the plane : = from r = to r = b A [. The 



* Phil. Trans, clxi. p. 77 (1870) ; Scientific Papers, vol. i. p. 33. Phil. 

 Mag. xliv. p. 328 (1872) ; Scientific Papers, vol. i. p. 140. Compare also 

 Phil Mag. xlvii. p. 566 (1809). xxii. p. 225 (1911). 



t The surface at z= -|-x may evidently be disregarded. 



