168 THE THEORY OF SCREWS. [167, 



Eliminating 6, and omitting the accents, we have, as the equation of the 

 cubic, 



yx 1 + mx 2 sin 2ij + l-y + 2lmx cos 2^ ml 2 sin 2?; = 0. 



The chord joining the two points and & has, as its equation, 

 mac cos (0 + ) [cos (2rj - - ) + cos (0 - )] + ly 



- ml sin (2vi-0- ) cos (0 + ) - ml cos (0 - ) sin (0 + ) = 0. 



If this be the chord joining the screws of equal pitch, then 



+ = 0, 

 and the equation reduces to 



mas (cos 2?7 + cos 20) + ly ml sin 2^ = 0. 



We thus see that this chord, which in the general section envelops a parabola, 

 now passes constantly through the fixed point 



x = 0, 



y = + m sin 2?;. 



This result could have been foreseen ; for, consider that screw on the cylin- 

 droid (and there must always be one) normal to the plane which it intersects 

 at a point P, any ray in the plane through P is perpendicular to this screw, 

 and, therefore, by a well-known property of the cylindroid, must intersect 

 the curve again in two points of equal pitch. This point P is, of course, 

 the point whose existence we have demonstrated above. 



168. Relation between Two Conjugate Screws of Inertia. 



We have found the relation between a pair of conjugate screws of inertia 

 so important in the dynamical part of the theory, that it is worth while to 

 investigate the properties of the chord joining two such points in the central 

 section. It can readily be shown that this chord must envelop a conic. 

 This conic and the point P on the cubic through which all reciprocal chords 

 will pass, will enable the impulsive screw, corresponding to any instantaneous 

 screw, to be immediately determined. For, draw through any point S that 

 tangent to the conic which gives S as one of the two conjugate screws of 

 inertia which must lie upon it ; let S be the other conjugate screw ; then 

 the chord PS will cut the cubic again in the required impulsive screw. 

 The two principal screws of inertia are found by drawing from P that 

 tangent to the conic which has not P as one of the two conjugate screws 

 of inertia. The two intersections of this tangent, with the cubic, are the 

 required principal screws of inertia. 



We can also determine the relation between the impulsive screw and the 

 instantaneous screw with regard to any section whatever. We have here 



