Eoever — Geometrical Constructions of Lines of Force. 203 



Equations (1) and (2) were obtained from electrical con- 

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

 be obtained from geometrical considerations. 



In Fig. 1 suppose A and A to be the traces of two axes of 



rotation, each perpendicular to the plane of the paper, and 



AP and AP two right lines in the plane of the paper. AP 



rotates around A with an angular velocity w^, and -4'P around 



m' 

 A with an angular velocity m'j, such that — ? = JS", a con- 



stant. If the lines rotate in the same direction (in the figure, 

 counter clockwise) and AP starts from a position AX and 

 ^P from a position AG^ making an angle a with AX^ the 

 locus of the point of intersection, P, is 



(o = — ? (cD — a ) , 



in which w = ^ PAX, a)' = ^ PAX and a = ^ GAX. 



The above equation written in another form is 



m\Q} — rWjO)' = m\a (3). 



When the rotations are in opposite directions the locus of 

 the point of intersection, P', is 



m\ 

 co' = — (a — cd) 



or 



in\(o 4- m^di = m\a (4). 



Equations (1) and (3) have the same form, but the primed 

 constants are interchanged. This shows that the curve repre- 

 senting a line of force proceeding from a system consisting of 

 two parallel electrified lines having charges of opposite alge- 

 braic sign is the locus of the intersection of two straight 

 lines rotating iu the same plane about the two parallel lines 

 as axes with angular velocities having the same algebraic sign 

 {i. e., the lines rotating in the same direction); also that 



