165] THE GEOMETRY OF THE CYLINDROID. 163 



The discriminant of the cubic 



- A n -H 3 + A CH* - SDH + D* = 

 is 



A -D* (A 2 D* + B 3 D + -fa AC 3 - &C* - f ABCD). 



Omitting the factor A~D 2 , we have, for the envelope of the system of cones, 

 the cone of the fourth order, found by equating the expression in the 

 bracket to zero. It may be noted that the same cone is the envelope of 

 the planes 



165. Application to the Plane Section. 



We next study the chord joining a pair of reciprocal points on the cubic 

 of 160. Take any point in the plane of the section ; then, as we have just 

 seen, a cone of screws can be drawn through this point, each ray of which 

 crosses two reciprocal screws. This cone is cut by the plane of section in 

 two lines, and, accordingly, we see that through any point in the plane of 

 section two chords can be drawn through a pair of reciprocal points. The 

 actual situation of these chords is found by drawing a pair of tangents to a 

 certain hyperbola. This will now be proved. 



The values of 6 and 6 , which correspond to a pair of reciprocal points, 

 fulfil the condition 



tan0tan0 =#; 



whence, 



cos (6 -0 ) = \ cos (0 + ) ; 



where, for brevity, we write X instead of 



l+H 

 l-H 



If, further, we make + 6 = ty, we shall find, for the equation of the chord, 



Px + Qy + Rz = 0; 

 in which, 



ra ,, . m . m 



P = -^ (X + cos 2a) -f sin 2a sin 2i/r + -- (\ + cos 2a) cos 2^, 



22 ^ 



Q = /t sin y3 + cos ft sin 2a ( (\ + cos 2a) sin 2\fr 



A & 



-~ cos 8 sin 2a cos 



7)1- 



= -h&quot; tan 8 - (\ 2 - 1 ) cot B + \hm sin 2-f - ~ (\ - 1) cot B cos 



11-2 



