162 THE THEORY OF SCREWS. [164, 



Let a, /3, 7 be the co-ordinates of the point. Then the plane through 

 this point, and the generator of the cylindroid defined by the equations 



y = x tan 0, 

 z = in sin 20, 

 is (y x tan 6) (7 m sin 20) = (/3 a tan 6) (z m sin 20) ; 



or, if we arrange in powers of tan 0, we obtain 



A tan 3 + B tan 2 + Otan + D = 0, 



in which 



A = VLZ yx ; D = &amp;lt;yy (3z, 



B = yy /3z + 2mx 2ma ; C = a.z yx + 2m/3 2my. 



If the same transversal also crosses the generator defined by , then, 



A tan 3 + B tan 2 & + G tan & + D = 0. 

 When the two screws defined by and are reciprocal, 



tan tan = H, 

 when H is a constant. 



By eliminating and rejecting the factor 



tan - tan & 



we obtain 



- A*H 3 + A CH* - BDH + D n - = 0. 



And as this is of the second degree in x, y, z, the required theorem has been 

 proved. 



All these cones must pass through the centre of the cylindroid, inasmuch 

 as the two principal screws of the cylindroid are reciprocal. If a constant 

 be added to the pitches of all the screws on the cylindroid, then the pairs 

 of reciprocals alter, inasmuch as H alters. The cone changes accordingly, 

 and thus there would be through each point a family of cones, all of which, 

 however, agree in having, as a generator, the ray from the vertex to the 

 centre of the cylindroid. Thus, even when the cylindroid is given, we must 

 further have the pitch of a stated screw given before the cone becomes 

 definite. This state of things may be contrasted with that presented by the 

 cone of reciprocal screws which may be drawn through a point. The latter 

 depends only upon the cylindroid itself, and is not altered if all the pitches 

 be modified by a constant increment. 



