186 Mr. A. Schuster on the Disruptive Discharge 



in which the quantity called y above is expanded in powers 

 of log q. If the spheres are near together so that q is nearly 

 equal to one, the higher powers of log q become sufficiently 

 small to be neglected. KirchhofFs formula is, however, very- 

 inconvenient for purposes of calculation, as the determination 

 of factors of the series is very complicated. It is of use only 

 when the spheres are so near together that the first two terms 

 are sufficient. 



I have transformed the equation so as to render it more 

 easily applicable to the problem I am discussing, and have 

 deduced a very simple expression which will give accurate 

 results, if the explosive distance is less than the fifth part of 

 the radius of the spheres. If the spheres are further apart, 

 the series (3) or Table I. must be used. 



Kirchhoff 's equations applied to the case of two equal 

 spheres are as follows: — 



B-Vy/a, (1) 



- (1 + g) 2 



V ~ 2(l- ? Kylog< ? n W 



n=u-iio g? ^- + r - 2 (io g? )^ 



" 1.2 8.4 (1 °gg )4 ^ •*•••• < 7 > 



The factors B are Bernoulli's numbers and 



U=l/(«* ■*«-*), 



?=flog? (8) 



In order to apply the equations to any special case, we 

 have first to obtain q from equation (2), f from equation (8), 



and next to find the numerical values of U, -7^-, —r-^ &c. The 



dg d£ 2 



labour of calculation is very great even for a few terms, 

 and it is found that the coefficients in the series (7) become 

 very large, so that many terms of the series are necessary 

 even for comparatively small distances between the spheres. 



I shall transform the equations in the first place by ex- 

 panding the differential coefficients of U in terms of log q. I 

 start from the equation (De Morgan, ' Differential and Inte- 

 gral Calculus,' p. 249) 



