Il6 PARTICULAR CASES OF EQUILIBRIUM. 



A; an analogous equation will give those which proceed from the 

 point A'. If m and m' are of the same sign, all these lines are 

 unlimited ; any one of them that of the order N, for instance is 

 an asymptote to a right line making an angle a with the axis Ax ; 

 this angle is denned by the condition that the right lines of order 

 n', connected by the ratio (4), are parallel to each other that is, 

 that we have 



n ri n + n' N m 



(6) a = = = - = - = - 0. 

 2m 2m 



All these asymptotes pass through the centre of gravity O of the 

 masses m and m' t which is evident and easy of verification. 



By eliminating the ratio , the equation of a line of force (5) 

 m 



and that of its asymptote give 



6-o> 0-a 



This is the equation of the line of force as a function of the 

 angles which the asymptote, and the tangent at the origin make with 

 the axis Ax. 



If r and r' are the distances of a point P to the two lines A and A', 

 the equation of the equipotential surfaces is 



V = const - 2 \M.r + X'l.r'} = const - 2/.(rV A/ ), 

 from which 



r \ r '\' _ C onst. 



134. SEVERAL PARALLEL LINES. It is evident that this method 

 of construction may be applied to any number of electrified lines 

 A, A', A" ... defined as above by the masses m, m' t m", . . . , on the 

 condition that these lines are parallel and situate in the same plane. 



The general equation of the lines of force starting from the 

 centre A of the mass m will be in this case 



the masses m, m',... may be positive or negative ; that of the cor- 

 responding asymptote is 



(m + m' + m" )a = md 



