Discharge of Electricity from Cylinders and Points. 797 



The effect of concentrating the force near an electrode 

 may be seen by considering the case in which the system of 

 forces between two conductors is slightly less than that 

 which is required to produce a discharge. If the integral 



f 



*e-L(°-®* ds 



along the path of the discharge is slightly less than unity, it 

 may be brought up to the required value by increasing the 

 force for a short distance ds either at the positive or at 

 the negative electrode. 



If a and /3 be increased to ct + 8^ and @ + 8j3i from s = 

 to s = Bs then the increase in the integral is Sfi^s. If a and 

 /3 be increased to a + £a 2 , and j3+8f3 2 from s = l — 8s to s = l 

 the increase in the integral is 



. -f («-P)ds 



os oa 2 e o 



Hence, it is more advantageous to increase the force at the 

 negative electrode than at the positive if 



'% exceeds **,-£<-»* 

 dX. AA 



For the case of a uniform field the condition for sparking is 



fi = ae ^' . 



Hence, if the uniform force is not great enough to produce 

 a discharge, it is more advantageous to increase the force at 

 the negative electrode than at the positive electrode if 



1 d/3 1 da 



fi dX. ol dX. 



The determinations of ct/p and fi/p that have been made for 

 the larger values of X/p, from X/p = 150 to 500, show that 



over this range the quantity -= -& is greater than - -j^ and 



the discharges are generally obtained more easily when the 

 force is concentrated at the negative electrode. The spark- 

 ing potentials are comparatively small in these cases, but as 

 X/p diminishes the sparking potentials increase and the effect 

 of concentrating the force at the negative electrode may be 

 quite different. 



It would, however, be impossible to find from theoretical 



