﻿834 Mr. G. B. Jeffery on Self-Intersecting 



Hence, § n = A sin (nd — a) 



and 



<f> = Ar n sin (lid— «) + terms in liigher powers of r. . (8) 



The tangents in the meridian plane to the branches of the 

 equipotential surface through P are in the directions 



_« ol-Vtt a+in — 1)tt 



**~n> ~^T' ' ~~n ' 



which agrees with Rankine's theorem, 

 <j) and -v/r are connected by the relations 



d£_lM d</> _ _ 1 ~df 

 which , by the aid of (6). transform into 



a<*> i b^ ib^>_ _i s^ 



By — r(a + r sin 0) B# ' r d# "~ a+r sin dr ' 



From these relations, together with (8 J, it is easy to 

 obtain 



^r= — Aar n cos(nd — a) + terms in liigher powers of r. (9) 



The tangents to the lines of force through P are therefore 

 in the directions 



2o + 7T 2a+37r 2cc+(2n-l)7r 



6 ~~ 2n ' 'In ' ' 2n ~" 



Hence, at a point of equilibrium not lying on the axis of 

 symmetry, the intersecting equipotential surfaces make 

 equal angles with each other; the lines of force also make 

 equal angles with each other, and bisect the angles between 

 the equipotential surfaces. 



If the point of equilibrium lies on the axis of symmetry, 

 these results are no longer true. The potential in the 

 neighbourhood of P can be expressed in terms of zonal 

 harmonics, 



= A/-"r,(/x) + B/-«+ 1 P ll+1 (/i)+ ...., . . (10) 



where /u, = cos 0. 



The tangents in the meridian plane to the equipotential 

 surfaces through P are, therefore, given by the roots of 



p»GO=o. 



This equation has n distinct roots between fi= +1 exclusive. 



