DYNAMICS OF A RIGID BODY. 1 2 1 



By taking the tangent planes to the pitch quadric at 

 the extremities of y y we should similarly find 



hence we deduce the very important result which may 

 be thus stated : 



The three principal axes of the pitch quadric, when fur- 

 nished with suitable pitches /, p& p y , constitute screws be- 

 longing to the screw complex of the third order, and the 

 equation of the pitch quadric has the form 



A* 2 + P$* + A 22 + AAA = - 



We can also show conversely that every screw of 

 zero pitch, which belongs to the screw complex of the 

 third order, must be one of the generators of the pitch 

 quadric. For 9 must be reciprocal to all the screws 

 of zero pitch on the reciprocal system of generators of 

 the pitch quadric; and since two screws of zero pitch 

 cannot be reciprocal unless they intersect either at a 

 finite or infinite distance, it follows that must inter- 

 sect the pitch quadric in an infinite number of points, 

 and must therefore be entirely contained thereon. 



Let now S denote a screw complex of the third order, 

 where a, /3, y are the three screws of the system on 

 the principal axes of the pitch quadric. Diminish the 

 pitches of all the screws of ,5* by any magnitude k. Then 

 the quadric 



must be the locus of screws of zero pitch in the altered 

 system, and therefore of pitch + k in the original system 



( so). 



