214 VI. SYSTEMS OF VECTORS. DISTEIBUT. OF MASS. INSTANT. MOTION. 



theorem, the point of intersection of the plane with CD is the focus 

 of the plane. Resolving at any point P in AB, the moment of OD, 



being perpendicular to OD 



E and OP, lies in the plane 



OAR 



That line in a plane which 

 has the property that for all 

 its points the resultant 

 moment lies in the plane is 

 called the characteristic of 

 the plane, or of its focus. 

 Its distance OX = d from the 

 focus is such that 1 ) 



14) dEsmfr = S. 

 The line OX, of length 



S 

 Rsinfr' 



d 



S 



is perpendicular to the plane 



of E anc( S, and drawn toward the side corresponding to the motion of 

 a right-handed screw when rotated in the direction from E to S. If 



we should go from in the direction OX a distance d' = ^ we 

 should reach the central axis, and 

 15) dd ' = W' 



63. Complex of Double -lines. If a plane 1 pass through 

 the pole of a plane 2, then the plane 2 passes through the pole of 



the plane 1. Let P (Fig. 64) 

 be the pole of the plane 1, and 

 let PO be any line in 1 through P. 

 The moment of E about is 

 perpendicular to PO, and so 

 is S, hence so is their resultant. 

 Thus the moment at is per- 

 pendicular to OP, and the polar 

 plane of contains the line OP, 

 that is, if 0, the pole of 2 lies 

 in 1, then P, the pole of 1 lies 

 in 2. 



In this case the two poles lie in the line of intersection 

 of the planes, and we see that if a plane turns about a line through 

 its pole, its pole traverses that line. Such a double line is conjugate 



Fig. 64. 



1) For the component in AS, E sin #, has the moment S about 0. 



