128 THE THEORY OF SCREWS. [138- 



If A and B be any two impulsive screws, and if A and B be the corre 

 sponding instantaneous screws, then the chords AR and BA will always 

 intersect upon the fixed right line XY. 



This right line is called the homographic axis. It intersects the circle in 

 two points, X and Y, which are the double points of the homographic systems. 

 These points enjoy a special dynamical significance. They are the two 

 Principal Screws of Inertia, and hence 



The homographic axis intersects the circle in two points, each of which 

 possesses the property, that an impulsive wrench administered on that screw will 

 make the body commence to move by twisting about the same screw. 



The method by which we have been conducted to the Principal Screws 

 of Inertia shows how there are in general two, and only two, of these screws 

 on the cylindroid. The homographic axis is the Pascal line, for the 

 Hexagon AA BB CC , and thus we have a dynamical significance for 

 Pascal s theorem. 



139. Determination of the Homographic Axis. 



The two principal screws of inertia must be reciprocal, and must also be 

 conjugate screws of inertia ( 84). The homographic axis must therefore 

 comply with the conditions thus prescribed. We have already shown ( 58) 

 the condition that two screws be reciprocal, and ( 135) the condition that 

 two screws be conjugate screws of inertia, and, accordingly, we see 



1. That the homographic axis must pass through 0, the pole of the 

 axis of pitch. 



2. That the homographic axis must pass through , the pole of the 

 axis of inertia. 



The points and having been already determined we have accordingly, 

 as the simplest construction for the homographic axis, the chord joining 

 and . 



140. Construction for Instantaneous Screws. 



The points and afford a simple construction for the instantaneous 

 screw, corresponding to a given impulsive screw. The construction depends 

 upon the following theorem ( 81): 



If two conjugate screws of inertia be regarded as instantaneous screws, then 

 the impulsive screw corresponding to either is reciprocal to the other. 



Let A be an impulsive screw (fig. 22); if we join AO we obtain H, the 

 screw reciprocal to A ; and if we join HO we obtain A , the conjugate screw 



