492 THE THEORY OF SCREWS. 



are the common tangents of a curve of the fourth and a curve of the second class. 

 Two and only two of these intersect the united lines of the involution. The 

 occurrence of the square factor indicates that the remaining six coincide in pairs 

 and hence we have the general result that the conic has triple contact with the 

 cubic. 



NOTE V. 

 Remarks on 210. 



Professor C. J. Joly has communicated to me the following theorems with regard 

 to the cubic which is the locus of the points corresponding to the screws of a 

 3-system which intersect a given screw of the system ( 210). 



Let be the double point on the cubic and P 1} P 2 the two points correspond 

 ing to a pair of screws of equal pitch which intersect 0. Then all the chords P^P^ 

 for different pitches are concurrent. 



For the cylindroid denned by the screws corresponding to P^ 2 must be cut by 

 the screw corresponding to in a third point which lies on the generator of the 

 cylindroid such that and are at right angles ( 22). As there is only one screw 

 of the 3-system intersecting at right angles it follows that all the chords PjP 2 

 will be concurrent. The point corresponding to is that whose co-ordinates are 

 given on p. 213, viz. 



Ps-Pz Pi-P* Pa- Pi 

 a l a a a :i 



where aj, a 2 , a 3 are the co-ordinates of 0. 



There is also to be noted the construction for the tangents at the double point 

 of the cubic. They are the lines to the points in which the pitch-conic through 

 the double point is met by the polar of the double point with respect to the conic 

 of infinite pitch. 



Let S p = be the conic of pitch p. Let P p - be the polar of with respect 

 to the conic of pitch p and let S p = be the result of substituting the co-ordinates 

 of in the equation of the conic S p = 0. 



As all the conies have four points common, suppose 



where k and I are certain constants. 



Likewise P =kP p + lP^- S ^kS p + lS x , 



whence after a few steps (210) we have the new form for the cubic 

 2WJS -?&amp;lt;W +SJSA=Q, 



If S p = passes through the double point then S p = and the cubic is 



