276 Trans. Acad. Sci. of St. Louis. 



If in equation (4) co = tt 



TT ' = Xtt + TTffX = Xtt + 'TTCrXf^. 

 Substituting this value of IT', equation (4) becomes 



7r<T (x — Xq) =X (tt — (o) (6). 



For o> = TT — co^ and a = ir — a equation (5) becomes 



\ (O)' a' ) = TTfTX, 



which is the same as equation (2). Also, for co = it — at 

 equation (6) becomes 



Tra (x — Xq) = \co\ 



which is the same as equation (3). This shows that the lines 

 of force proceeding from a system consisting of an electrified 

 plane and an electrified line parallel to the plane, are curves 

 of the same kind whether the charges are of like or of unlike 

 signs. 



Since the force at any point due to an electrified plane is 

 independent of the distance of the point from the plane, it 

 follows that the lines of force of the above system are 

 the same regardless of the distance of the electrified line from 

 the electrified plane. 



The preceding equations were obtained from electrical con- 

 siderations. In what follows it will be shown how they can 

 be obtained from geometrical considerations. 



In Fig. 1. is the trace of an axis of rotation which is 



perpendicular to the plane of the paper ; OP and DP are 



two straight lines in the plane of the paper. OP rotates 



about O with a uniform angular velocity a and PD moves in 



a direction perpendicular to itself, which direction is parallel 



to ABf with a uniform linear velocity v. If OP rotates 



, . ri ' ( right handed > J. .. i t->ti 



about U m a.< '^ S> direction and IJP moves to 



C left handed > 



the < ,*„ S , and if PZ) has a position OY when OP has 

 ( left S ^ 



