181] FREEDOM OF THE THIRD ORDER. 183 



lines. One of these lines is the required residence of the screw 0, while 

 the other line, with a pitch equal in magnitude to that of 0, but opposite 

 in sign, belonging, as it does, to one of the other system of generators, is a 

 screw reciprocal to the system. 



The family of quadric surfaces of constant pitch have the same planes 

 of circular section, and therefore every plane through the centre cuts the 

 quadrics in a system of conies having the same directions of axes. 



The cylindroid which contains all the screws of the screw system parallel 

 to one of the planes of circular section must be composed of screws of equal 

 pitch. A cylindroid in this case reduces to a plane pencil of rays passing 

 through a point. We thus have two points situated upon a principal axis 

 of the pitch quadric, through each of which a plane pencil of screws can be 

 drawn, which belong to the screw system. All the screws passing through 

 either of these points have equal pitch. The pitches of the two pencils are 

 equal in magnitude, but opposite in sign. The magnitude is that of the 

 pitch of the screw situated on the principal axis of the pitch quadric*. 



181. Virtual Coefficients. 



Let p be a screw of the screw system which makes angles whose cosines 

 are /, a, 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 p, 



&c., &c., 



ps =/e +g@6 + hy6 

 Let ij be any given screw. The virtual coefficient of p and rj is 



Draw from the centre of the pitch quadric a radius vector r parallel to p, 

 and equal to the virtual coefficient just written ; then the locus of the 

 extremity of r is the sphere 



x 2 + \f + z* = #CT ar) + yet ft + zvr yrl . 



The tangent plane to the sphere obtained by equating the right-hand 

 side of this equation to zero is the principal plane of that cylindroid which 

 contains all the screws of the screw system which are reciprocal to 17. 



* If a, b, e be the three semiaxes of the pitch quadric, and +d the distances from the centre, 

 on a, of the two points in question, it appears from 179 that 2 d 2 = (a 8 -i 2 ) (a 2 -c 2 ), which shows 

 that d is the fourth proportional to the primary semiaxis of the surface, and to those of its focal 

 ellipse and hyperbola. 



