s+^ + ^+W=0. •■•••, • (3) 



Rain-Clouds and Atmospheric Electricity. 199 



is an imaginary charge p' snch as would produce the actual 

 distribution of potential that we have in the real case, then 

 o^Y cPY cfj 

 dec 2 dy 2 dz 

 From equations (2) and (3) we find 



4K'-— •— — .— .^.^1. 

 ^ ~~ dx dx dy dy dz dz ' 



or if p is the density per unit volume of a real charge at any 

 point of the non-homogeneous dielectric, then 



4 K / -4tt + — .— + —. — + ^5.^Y . 



^ ~~ - dx dx dy dy dz dz 



Also if at any place there is a distinct separation by a sur- 

 face of one dielectric from another, ordinary air from very 

 moist air for example, then the resultant force on one side of 

 the surface must be greater than that on the other. Thus, if 

 the resolved part of the resultant force in a direction at right 

 angles to this surface be F in the first medium and F' in the 

 second, and if K and K7 are the specific inductive capacities, 



KFzrrK'F'. 



In fact it is the same as if both media were dry air as above, 

 and an apparent charge, of density </, were given to the sur- 

 face, such that 



47r^=(l+^)F 



-.£-0 



If at the surface there is a real charge of density cr, then 



KF = K'F' + 4iro-; 

 and the action is as if both media were air as above, and an 

 apparent charge of density a* were given to the surface, where 



-£ + (i-$)»: 



These formulae may be used, when we know the state of the 

 atmosphere at every point at every time (that is, when the 

 specific inductive capacity K is known), in order to find from 

 any given initial distribution of potential the changes Which*- 

 occur when the state of the atmosphere is changed. 



It is known that, from observations of atmospheric elec- 

 tricity and from observations of earth-currents,- atmospheric 



