Discharge of Electricity from Points. 241 



and focus. The effect o£ the plane may be represented by 

 adding the potential of the image in the plane of the 

 paraboloid. 



The total potential is thus 



V=Alogr(l + cos#)-AlogV(l+ cos0'). 



The diameter of a small point may be taken as the breadth 

 of the wire at the point where the breadth is twice the 

 distance from the vertex*. 



Hence the parabola whose latus rectum is 21 corresponds 

 to the point whose diameter 2a equals 4Z. 



This parabola passes through the point 

 Along the axis 



V=Alog^=Alog 



I a 



r= a = I 



and A is given by V = V 1 , when r=a/4z. 



.'. Y 1 = Alog7- / approximately, since j is small. 



If X is the force at any point along the axis 



A -A 



X=- 



r 



hta 



where #, the distance from the vertex, is small. 



4A 



X x (the force at the vertex) =■ . 



It is now possible to find an approximate formula for the 

 sparking potential by the method used by Townsend for 

 cylinders!. It is known that practically no ionization by 

 collision takes place in air at atmospheric pressure if the 

 force is less than 30 kilovolts per centimetre. Also Bailie's 

 results t give the empirical formula V== 30 s + 1' 35 for the 

 sparking potential V in kilovolts through air at atmospheric 

 pressure when s, the length of spark-gap, is of the order of 

 a millimetre.- 



Let s be the distance along the axis from the vertex of the 



* This was the method adopted by J. Zeleiry, Phys. Rev. vol. xxvi. 

 p. 130 (1908). 



t J. S. Townsend, ' Electrician,' June 6, 1913. 



X Bailie, Ann. de Chim. et Thys. [5] vol. xxv. p. 486 (1882). 



Phil. Mag. S. 6. Vol. 2S. No. 164. Aug. 1914, R 



