182 Prof. D. N. Mallik on Lines of 



The latter can be shown to pass through the point of 

 equilibrium on the axis and to separate the two sets of lines 

 which issue from the points A and B. In the fig. III., 



q = 15, q' = 5, 



and r + s = 10 

 separates the two sets of lines. 



8. Case V. — A charge e in a uniform field. 

 From the equation 



e cos — e' cos 6' = constant, 



we have 



e' 

 ecos# + -^r 2 (l — cos 6') = constant, 



where r is the distance of any point P on a line of force 

 from e'. 

 This gives, 



e cos + -g -5- +..... = constant. 



Now, when r is infinite, rO = y, 



y being the ordinate of the point P, with AB as the axis of a, 

 drawn parallel to the direction of the lines of force due to 



the uniform field and -= = — X, where X is the uniform force 

 r 



acting in the field, measured along AB. 



The equation thus becomes 



e cos 6 ~ — constant. 



9. This can also be obtained directly from the differential 

 equation : thus, if R, 6 are the polar coordinates of any 

 point P, we have 



1,e *' ~d0 + Xdy = O. 



But since y = B sin 6, we have 

 e sin QdQ + Xy dy = 0, 



1. e., e cos ~ = constant. 



z 



