212 VI- SYSTEMS OF VECTORS. DISTBJBUT. OF MASS. INSTANT. MOTION. 



the focus of the plane. Let the plane cut the central axis in A. 

 Through A draw a plane perpendicular to the central axis, intersecting 

 the given plane in AO. As we go along the line AO, S turns 



about it, and for one point 

 has the direction of the 

 normal to the given plane. 

 Accordingly to every 

 point in space there corre- 

 sponds one plane, and to 

 every plane one point. 

 The correspondence was 

 discovered by Chasles, and 

 the system of points and 

 planes was called a Null- 



Fig. 60. 



System by Mobius. 



62. Vector -cross. Besides the reduction to the screw -type 

 we may reduce the system of vectors to two vectors not lying in 

 the same plane, without a couple. This reduction may be made in 

 an infinite number of ways, and the line of one of the vectors may 



Fig. 61. 



be given. Let AB (Fig. 61) be the given line. At any point 

 on AB let E be the resultant vector, S the resultant couple. 

 (E and 8 will not in general lie in a plane with AB.) At pass 

 a plane perpendicular to S, intersecting the plane of E and AB 

 in OP. Resolve 8 into the pair of vectors OP and CQ so taken 

 that the resultant of E and OP shall lie in AB. The length of OP 

 is thus determined, and the distance between its line and that of CQ 

 is determined by S. Thus the line AB determines the line CQ. 



