of Electricity through Gases. 183 



second part of the paper I have discussed the discharge from 

 the theoretical point of view, and although no very definite 

 results are arrived at, owing to our ignorance of the properties 

 of a gas in the layer which is in contact with the solid, I 

 hope the discussion may serve to point out the direction in 

 which further experimentation is at present desirable. 



Part I. — Calculation of the Disruptive Tension at the Surface 

 of a Sphere. 



Whatever idea we may form of the nature of a disruptive 

 discharge, it seems natural to suppose that the spark passes when 

 the normal force at some point of a conductor has reached a 

 certain value. If the discharge takes place between two equal 

 spheres, the point for which the normal force R has to be cal- 

 culated is on the line joining the centres, where R has its 

 greatest value, for it is there that the spark will pass. I shall 

 use the equations in the form given them by Kirchhoff^; and 

 treat of the case only in which one of the spheres is at poten- 

 tial zero : the problem then consists in finding the potential 

 of the other sphere when the normal force, where it is greatest, 

 has reached a certain value. 



If a is the radius of the spheres, ac the distance between 

 the centres, R the normal force at that point of the sphere 

 having a potential V which is nearest to the sphere of zero 

 potential, we have the equation 



R=Yyja, ....... (1) 



where y is a function of c. 

 Writing 



2q=c- \/c 2 -4, (2) 



TO — 1 



~2~ 



M)— X _ q 2 1 _ q e^ 1 __ q io 1_^ + ...±1^2to,. W 

 we obtain easily from KirchhofFs equations, 



y-.^f/b) ••••••• (4) 



When the spheres are near each other so that c is nearly 

 equal to 2, and therefore q is nearly one, the series (3) con- 

 verges very slowly, and the potential has to be calculated in 



* Collected Works, p. 78. There is a misprint in the equation for 

 d-JJ/dC 2 on p. 99. The factor on the right-hand side of the equation should 

 be 4 instead of 3. 



