216 VI. SYSTEMS OF VECTORS. DISTRIBUT. OF MASS. INSTANT. MOTION. 



or double lines are lines of no moment. In the kinematical applica- 

 tion, points on a double -line experience no translation along it. 



If a double -line cuts one of a pair of conjugate lines, it cuts 

 the other. Let PQ be a double -line cutting the line AS. Then 

 the pole of the plane BPQ lies in the line conjugate to AS. But 

 since PQ is a double-line, the pole of BPQ lies on PQ. Hence 

 PQ cuts the conjugate to AB. Conversely, every line cutting two 

 conjugates is a double -line. 



The complex of double -lines is symmetrical with respect to the 

 central axis. Let AB (Fig. 65) be a line of the complex, and let OX 



be the common perpendicular to it and 

 the central axis. Now AB is perpen- 

 dicular to the moment S at X, but S 

 is perpendicular to OX, and the distance 



o 



OX is d = -jr tan #-. If <p is the angle 



that the line AB makes with the central 

 axis we have 



19) 





This equation shows that the double-lines 

 constituting the complex are tangent to 



an infinite number of hjelices, which become less steep as d decreases, 

 so that the double -lines cutting the central axis are perpendicular to 

 it ; and those at infinity are parallel to it. For the pitch p of any 

 helix tangent to lines of the complex we have 



20) 



p 







Ed 



Thus the pitch is not constant, but varies as d 2 . 



This construction shows the triple infinity of complex -lines. In 

 a plane perpendicular to the central axis every point is on one 

 complex line. There is a double infinity of such points. But there 

 is a single infinity of such planes, and therefore in all a triple infinity 

 of complex lines. It is evident that the complex is unchanged if 

 we rotate it about, or slide it along the axis. 



64. Composition of Screws. Suppose we have two systems 

 of vectors, each reduced to the type of a screw. (The combination 

 of forces of this type, namely a force, and a couple tending to cause 

 rotation about its line of direction, is called a wrench. E is called 

 the intensity of the wrench, or the amplitude of the rotation.) The 

 resultant of both systems may also be reduced to a screw, and we 

 may find its position. 



