202] PLANE REPRESENTATION OF THE THIRD ORDER. 201 



We have now to investigate the locus of the screws of given pitch, and as 

 p is presumed to be a determinate quantity, we have 



(p.* - p) 2 + x0 3 = 0, 



whence, by eliminating 1} 2 , 3 we obtain, as the locus of the screws of 

 pitch p, the quadric otherwise found in the previous chapter 



(p! -p) x z + (p,-p) y + (p.-p) z- + (pi-p)(p s -p)(p 3 -p) = 0. 



According as p varies, this family of quadrics will exhibit all the screws of 

 the three-system which possess a definite pitch. 



202. Imaginary Screws. 



To complete the inventory of the screws it is, however, necessary to 

 add those of indefinite pitch, i.e. those whose co-ordinates satisfy both the 

 equations 



M +M a +M 8 =o. 



0s + e.? + &amp;lt;v=o. 



There are four triads of co-ordinates which satisfy these conditions, and, 

 remembering that only the ratios are concerned, the values of 1} n , 3 

 may be written thus : 



The equations of the axis written without p are 



* (Of + 0J} - yOA - ^0 A + ( P* - p 3 ) 0,6, = 0, 

 y (Of + 0s) - zOA - x6A + (p 3 - Pl ) 6A = o, 



z (6? + &amp;lt;9 2 ) - x0A - y0 3 0, + ( PI - p z ) 6,0, = 0, 

 of which two are independent. 



If we substitute the values of 0,, 6.,, 3 for the first indeterminate screw, 

 the three equations just written reduce to 



* ( P2 - #,)* + y ( p 3 - p$ + z ( Pl - p$ -(p 2 - p 3 ) h - (p-.-p^^p,- p$ - 0. 



