GEOMETRICAL PROPERTIES OF THE LINES OF FORCE PROCEEDING FROM (a) A SYSTEM CONSISTING OF AN ELECTRI- (b) A SYSTEM CONSISTING OF AN ELECTRI- POINT.* Wм. H. ROEVER. (a) The curve representing a line of force proceeding from a system consisting of an electrified plane and an electrified line parallel to the plane, is the locus of the intersection of two straight lines having motions in a plane which is perpendicular to the electrified line; one line having a motion of uniform rotation about the electrified line as an axis, and the other a motion of uniform translation perpendicular to itself and parallel to the electrified plane. The force at a distance from an electrified straight line The flow of force whose charge is λ per unit length is f 2λ 2λ = from a unit length of line is Χ 2πη = 4πλ. Hence the flow of force from a unit length of line between two planes which intersect in the line and make an angle o with each other is The force at a finite distance from an infinite electrified plane whose charge is a per unit area is F= 2πσ. Hence the flow * Read before The Academy of Science of St. Louis, December 7, 1896. of force from a rectangular portion of this plane whose length is x and whose width is unity is M = 2πσx. In Fig. 1. let O be the trace of an electrified straight line and AB the trace of an electrified plane, both of which are perpendicular to the plane of the paper. Through the line whose trace is O pass two planes, whose traces are OY and OP respectively; OY is perpendicD DB ular to AB and OP makes an angle = <YOP with OY. Also at a distance x = O'D from Opass a plane whose trace is DP, which is perpendicular to AB. The flow of force through the angle w from a unit length of the electrified line is A Ο' FIG. 1. Ν = 2λω, in which is the charge per unit length of the electrified line whose trace is 0. The flow of force through the rectangular prism determined by the planes OY and DP, and two planes perpendicular to the line O and at a unit's distance apart, is Μ = 2πσα, in which is the charge per unit area of the electrified plane whose trace is AB. Then if the line and plane have charges of unlike signs, the flow of force between the plane OY and the line of intersection P of the planes OP and DP is N-M=2x-2πσx. If in this equation we make N-M constant we confine P to a certain path. This path is the right section of a cylindrical surface which bounds a tube of force. Hence the locus of P is a line of force whose equation is in which K is a constant. If in equation (1) x = 0 Κ = λω = λα, in which a is the special value of w for which x = 0. stituting this value of K, equation (1) becomes Sub which is the equation of a line of force whose direction at O makes an angle a with OY. If in equation (1) w = 0 Substituting this value of K, equation (1) becomes Substituting this value of K', equation (4) becomes If in equation (4) w = π Κ' = λπ + που = λπ + πσ· Substituting this value of K', equation (4) becomes (« — £) x = (x —x) D (6). For wπ w' and a = Ta' equation (5) becomes П λ (w' — α' ) = πOX, which is the same as equation (2). Also, for w= equation (6) becomes 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 considerations. In what follows it will be shown how they can be obtained from geometrical considerations. In Fig. 1. O 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 AB, with a uniform linear velocity v. If OP rotates S right handed about O in a the {right} left handed direction and DP moves to tion and if PD has a position OY when OP has |