61, 62] NULL-SYSTEM. VECTOR-CROSS. 213 



The lengths OB and CQ are determined as soon as the line AB is 

 given. Two such non- parallel and non-coplanar vectors OB, CQ will 

 be termed a vector -cross. The crossing will degenerate to intersection 

 only when S = and to parallelism when R = 0. 



As any line may be taken for AB, and as there are a quadruple 

 infinity of lines in space, there are a quadruple infinity of vector- 

 crosses. They all possess a property in common, namely, that the 

 tetrahedron formed by joining the four ends of a vector- cross has a 

 constant volume. Let OB, CQ (Fig. 61) be the vector- cross, and let 

 us reverse the preceding resolution. The volume of a tetrahedron is 

 equal to one -third the product of its altitude by the area of its 

 base. The area of the base OCQ is one-half the moment of CQ 



about 0, or --- S, while the altitude is the projection of OB on the 



perpendicular to OCQ, that is, on S. But since BE is parallel to 

 the plane OCQ, OR has the same projection on S as OB, namely 

 .Rcos'fr, consequently 



v= 4^ cos #-4-#=4-^cos#. 



O 4 U 



But by 8), 



$COS# = S Q , 



therefore 



13) V=RS Q . 



This theorem is due to Chasles. 



Corresponding lines of vector -crosses possess a remarkable relation 

 to the null-system. Let AB and CQ (Fig. 62) be the two lines of 

 the vector- cross. Through CQ pass any 

 plane, cutting AB in 0. The moment 

 of CQ is perpendicular to the plane OCQ, 

 and the other vector has no moment 

 about 0, since it passes through it. Accord- 

 ingly is the focus of the plane OCQ. 

 Thus, if a plane turns about a line, its 

 focus traverses another line, and these 

 two conjugate lines are lines of a vector- 

 cross. 



We have here shown the intermediate 

 nature of a line between a point and a 



plane, in the dual role as generated by the motion of a point and 

 by the rotation of a plane. In the first relation the line is spoken 

 of as a ray, in the second as an axis. 



If two conjugate lines are at right angles, pass a plane through 

 one, AB, perpendicular to the other, CD (Fig. 63). By the preceding 



