Roever — Geometrical Constructions of Lines of Force. 217 



The clashed line (Fig. 3) is the limiting line, and separates 

 the region having branches passing through A, from that 

 having branches passing through A'. 



The angle which the asymptote to the limiting line makes 

 with OA is by equation (13) 



TT =: TT TT, 



m + in' m + tit 



m' 



in which — , '^ is the angle which the asymptotes to a 



m + m 



branch passing through A' , make with each other. 



(6) TJie curve representing a line of force proceeding from 

 a system consisting of two electrified points^ is the locus of the 

 intersection of tioo straight lines rotating in the same plane 

 about parallel axes, passing through those points, in such d 

 manner that the versines of their angles of inclination to the 

 plane of the axes change at uniform but diferent rates. 



Suppose A and A' (Fig. 1) to be the centres of two small 

 spherical conductors having charges -\-m and — m' , respec- 

 tivelv. The number of lines of force leaving: the mass m 

 through the circular cone whose vertex is at A and whose 

 semi-angle is ta is 



N = 27rm (1 — cos w ) . 



The number of lines of force conversrino; to — m' throuorh 

 the circular cone whose vertex is at A' and whose semi-anirle 

 is a>' is 



ISF' = ^irm' (1 — cos «'). 



The number of lines proceeding to the right through the 

 circle of intersection of the two co-axial cones is 



iV— iV' = 27rm (1— cosoi) — 2'irni' {\ — cos w'). 



The locus of all such circles of intersection is a tube of 

 force, and the meridian curve of such a tube must be a line 

 of force. If in the above equation &>' = 



N — N ' ■=■ 27rm (1 — cos (o) = lirm (1 — cos a). 



