288 Trans. Acad. Sci. of St. Louis. 



For versin « = 2 — versin to' and versin a = 2 — versin a' 

 equation (29) becomes 



m (versin a>' — versin a') = irax^f 



which is the same as equation (26). Also, for versin© 

 = 2 — versin co' equation (30) becomes 



Tra {x^ — x^) = m versin a>', 



which is the same as equation (27). This shows that the lines 

 of force proceeding from a system consisting of an electrified 

 plane and an electrified point are curves of the same kind 

 whether the changes are of like or of unlike signs. (Figs. 1 

 and 2.) In this case also the lines of force are the same 

 regardless of the distance of the electrified point 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. the straight line OP rotates about O and the 

 straight line PD moves in a direction perpendicular to itself, 

 which direction is parallel to AB. OP rotates about in 

 such a manner that the versine of the angle co, which it makes 

 with or" (a perpendicular to AB, through O), changes at a 

 uniform rate. This motion could be brought about mechani- 

 cally by making OP the center line or axis of a crank to be 

 driven by a slotted crosshead, which has a slot perpendicular 

 to and a uniform linear motion parallel to OY. Let s denote 

 the length of stroke of the crosshead and u the portion of that 

 stroke through which the crosshead moves in a unit of time. 

 Then su is the velocity of the crosshead. PD moves in such 

 a manner that if it were the meridian line of a circular cylinder 

 whose axis is OY, the area of cross section of this cylinder 

 would change at a uniform rate. Let ^ = nrv^ denote the 

 increment by which this area changes in a unit of time. 

 Then if PD starts from a position of coincidence with OY, 

 its distance from OY at the end of a unit of time will be 



x = v. Now if OP rotates in a ^ ° , , > direction and 



lett handed 



