176 ON FARADAY'S LINES OF FORCE. 



be proportional to the resultant attraction of all the particles. Now we may find 

 the resultant pressure at any point by adding the pressures due to the given 

 sources, and therefore we may find the resultant velocity in a given direction 

 from the rate of decrease of pressure in that direction, and this will be 

 proportional to the resultant attraction of the particles resolved in that direction. 



Since the resultant attraction in the electrical problem is proportional to 

 the decrease of pressure in the imaginary problem, and since we may select 

 any values for the constants in the imaginary problem, we may assume that the 

 resultant attraction in any direction is numerically equal to the decrease of 

 pressure in that direction, or 



dp 

 A - ~dx- 



By this assumption we find that if V be the potential, 



d V= Xdx + Ydy + Zdz = - dp, 



or since at an infinite distance F=0 and p = Q, V= p. 

 In the electrical problem we have 



In the fluid p = Z ; 



^j- 



tC 



If k be supposed very great, the amount of fluid produced by each source 

 in order to keep up the pressures will be very small. 



The potential of any system of electricity on itself will be 



, . 



If (dm), 2 (dm') be two systems of electrical particles and p, p' the potentials 

 due to them respectively, then by (32) 





or the potential of the first system on the second is equal to that of the second 

 system on the first. 



