Ball — Dynamical Problems in the Theory of a Rigid Body. 33 



fulfil one special condition — it must always pass througli 0, the pole 

 of the line by which the pitches are determined. This is easily shown, 

 for we have proved (" Screws," p. 48), that if Pj and Q3 be reciprocal, 

 then Pi and Qi_ must also be reciprocal. ISTow Pi may be chosen arbi- 

 trarily, and so can Q^, and, if reciprocal, the chord must pass through 

 ; therefore P^, Qi must pass through 0, or must be a point on the 

 axis of homography. The fact that the axis must pass through in- 

 volves as a consequence the theorem otherwise well known, that the 

 principal screws of inertia on the cylindroid are reciprocal. It is thus 

 interesting to note how the dynamical conception of conjugate screws 

 of inertia is illustrated by an elegant geometrical theory. 



"We can still further simplify the subject geometrically by the pro- 

 perty of the conjugate screws of inertia. Let A and £ be two fixed 

 points on a circle, and P and Q be two variable points ; then if a and /S 

 be both constants, and if the condition 



a sin PPA . sin QPA + j3 sin PAP . sin QAP = 



be fulfilled, then it is easy to show that the chord PQ must pass 

 through a fixed point ; but the condition that P and Q shall corre- 

 spond with a pair of conjugate screws is of this type (" Screws," p. 48), 

 and hence we have the interesting result that all the chords joining a 

 pair of conjugate screws of inertia, P and Q, must intersect in. the 

 same point 0'. 



By the help of this theorem we are now enabled to construct the 

 pairs of impulsive and instantaneous screws with the greatest facility. 



Fi£ 



Let P (Fig. 2) be an instantaneous screw ; then if we draw the chord 

 PO' it will intersect the circle again, at >S'; and then the chord SO 



U. I. A. PliOC, SEE, II., VOL IV. — SCIENCE. 



