130 DYNAMICS OF A RIGID BODY. 



through each of which a plane pencil of screws can be 

 drawn, which belong to the screw complex. All the 

 screws passing through either of these points have equal 

 pitch. The pitches of the two pencils are equal in mag- 

 nitude, but opposite in sign. The magnitude is that of 

 the pitch of the screw situated on the primary axis of 

 the pitch quadric.* 



1 1 6. Virtual Co-efficients. Let p be a screw of the 

 screw complex which makes angles whose cosines are 

 f, g, h, with the three screws of reference a, /3, y upon 

 the axes of the pitch quadric. Then, reference being 

 made to any six co-reciprocals, we have for the co- 

 ordinates of /o, 



&c., &c., 



Let j be any given screw. The virtual co-efficient of 

 and i is 



Draw from the centre of the pitch quadric a radius vec- 

 tor r parallel to /o, and equal to the virtual coefficient 

 just written ; then the locus of the extremity of r is the 

 sphere 



= 2 



The tangent plane to the sphere obtained by equating 

 the right-hand side of this equation to zero is the prin- 



* If a, 5, c be the three semiaxes of the pitch quadric, and + d the distances 

 from the centre, on <z, of the two points in question, it appears from 114 that 

 2<# = (a? - 2) (<z 2 - 2 ), which shows that d is the fourth proportional to the 

 primary semiaxis of the surface, and of its focal ellipse and hyperbola. 



