TWO PARALLEL LINES. 115 



This flow on the other hand is equal to N units ; we have therefore 



N 

 (3) Q = 2W0 = N, or (9 = . 



2M 



133. Two PARALLEL LINES. Suppose that the electrical system 

 consists of two parallel lines A and A', of densities A. and A/, such 

 that m = t\ /rc' = eA/; take for the x axis the straight line joining 

 the two lines A and A' (Fig. 28). 



Through these two points draw two straight lines An and A'n' 

 of the orders n and ri respectively in reference to the centres A 

 and A', and making angles w and o/ with the axis ; join their point 

 of intersection P to the axis by any given curve PP'. It is evident 

 that across the cylindrical surface PP', there is a flow , or 2mu from 

 A, and a flow n' from A', and therefore a total flow equal to n + n' = N. 

 The same will be the case with all the points of the curve AP, de- 

 fined by the points of intersection, two by two, of the straight lines 

 proceeding from A and A', and such that the sum of their numbers 

 is equal to N ; the locus of all these points is evidently a line of 

 force of the order N for the resulting system. 



The force near one of the acting masses depends only on this 

 mass, the influence of which predominates. The line of flow of the 

 order N is therefore tangential at A to the right line of order N 

 drawn from this point. 



The equation of the curve AP is 



(4) 



from which, replacing these quantities by their values as functions 

 of the angles, 



(5) mot + m'<i>' = mO. 



This is the equation of the lines of force drawn from the point 



I 2 



