208] 



PLANE REPRESENTATION OF THE THIRD ORDER. 



207 



A pair of reciprocal screws are conjugate points with respect to the zero- 

 pitch conic. 



From this theorem we can at once draw the following conclusions : 



All the screws of the system reciprocal to a given screw lie upon a 

 cylindroid. 



For the locus of points conjugate to 9 is, of course, the polar of 

 with respect to the zero-pitch conic, and this polar will correspond to a 

 cylindroid. 



On any cylindroid one screw can always be found reciprocal to a given 

 screw 9. For this will be the intersection of the polar of 9 with the line 

 corresponding to the given cylindroid. 



A triad of co-reciprocal screws will correspond to a self-conjugate triangle 

 of the conic of zero-pitch. 



208. The Principal Screws of the System. 



Draw the conic of zero-pitch A, and the conic of infinite pitch B, which 

 intersect in the four screws of indeterminate pitch, P,, P 2 , P 3 , P 4 (see fig. 39). 

 Draw the diagonals of the complete quadrilateral, and let them intersect in 



Fig. 39. 



the points X, Y, Z. These three points are significant. Take any pair of 

 them, X and F; then, by the known properties of conies, X and Y are con 

 jugate points with respect to both of the conies A and B. The screws X 



