290 Trans. Acad. Sci. of St. Louis. 



G'Y', then the locus of the point of intersection is expressed 

 by the equation 



2u (x^ — Xq^ ) = v^ (2 — versin cu ) ( 35 ) . 



Equations (34) and (35) may be simultaneously expressed in 

 the general form 



v"^ versin co + '2ux^ = E\ (36), 



in which K\ is a constant. 



Equations (33) and (36) represent identical curves. 



Equations (26), (27), (29), (30) have the same forms as 

 equations (31), (32), (34), (35) respectively, and this 

 shows: — 



I, That the curve representing a line of force proceeding 

 from a system consisting of an electrified plane and an elec- 

 trified point, is the locus of the intersection of two straight 

 lines having motions in a plane which passes through the 

 electrified point and is perpendicular to the electrified plane ; 

 one line having a motion of rotation about the electrified 

 point and the other a motion of translation perpendicular to 

 itself and parallel to the electrified plane. The rotation is 

 such that the versine of the angle which the rotating line 

 makes with OY (a. line which passes through the electrified 

 point and is perpendicular to the electrified plane) changes at 

 a uniform rate, and the translation is such that if the moving 

 line were the meridian line of a cylinder of revolution whose 

 axis is OY, the area of cross section of the cylinder would 

 chansfe at a uniform rate. 



'S3 



II. That the constants <7, m, v and u are related by the 

 equation 



^ ""' ' (37). 



<r 2u 2m 



III. That if the electrified point be considered as being 

 above the electrified plane 



