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LXXXYT. On Self-Intersecting Lines of Force and Equi- 

 potential Surfaces. By G. B. Jeffery, M.A., B.Sc, 

 Assistant in the Department of Applied Mathematics, Uni- 

 versity College, London *. 



BY a well-known theorem due to Rankine, if n sheets of 

 an equipotential surface intersect at a point o£ equi- 

 librium, they make equal angles ir/n with each other. The 

 object of this paper is to give some simple extensions of this 

 theorem and some analogous results for lines of force. We 

 will confine our attention to the cases in which the lines of 

 force can be defined in terms of a force-function, i. e. when 

 the field is either two-dimensional or has an axis of symmetry. 

 The case of a two-dimensional field admits of a very 

 simple treatment. Both the potential cf> and the force- 

 function ty satisfy the differential equation 



S+|?= CO 



Taking the point of equilibrium in question as origin, the 

 potential in its neighbourhood can be expressed in the form 



= H K + H M+ i + H n+2 +...., . . . (2) 



where H n is a homogeneous function of the coordinates of 

 degree n. Each term of this series must be a solution of 

 (1), and hence, in polar coordinates, 



K n =Ar n sm(n0-a) (3) 



The tangents to the branches of the equipotential surface 

 at the origin are given by the roots of H n = 0, L e. by 



a a + 7T a+27T a + (n— V)7T 



u= — , ? ~~ 5 ....... • 



n n n n 



Hence the n sheets of the equipotential surface make 

 equal angles irjn with each other. It is obvious that in this 

 case a precisely similar theorem holds for lines of forcef. 

 It remains to find the relation between the lines of force and 

 the equipotential surfaces at the same point of equilibrium. 

 6 and ^ are connected by the relations 



B_^_l^ l^4>_ _M 



dr~ra0' rW~ dr ' 



* Communicated by the Author. 



+ A proof of this property for any curve defined by a solution of (1) 

 ■was given by Stokes in a note appended to liankine's paper, Proc. R. S. 

 xv. 1867. 



