337] VARIOUS EXERCISES. 365 



and similarly, 



COS (yf;) = --,- _ COS 



COS ( 7 ) 



Pv i v\ P- 



/ _ cos(a) = . N cos 



cos ( 7 ) cos (a?;) 



whence we obtain 



cos (a) cos (fir)) cos ( 7 f ) + cos (a) cos (/3) cos (yrf) = 0, 



for it is shown in 283 that p a -:- cos (a??) or the other similar expressions can 

 never be zero. 



336. Instantaneous Screw of Zero Pitch. 



Let a be an instantaneous screw of zero pitch. Let two of the canonical 

 co-reciprocals lie on a, then the co-ordinates of a are 



i, i o, o, o, o. 



The co-ordinates of the impulsive screw ij are given by the formulae of 

 32G which show that 



We thus have 



(i + Oa) (Vi + fc) + ( 3 + 04) (ife + ^4) + ( 5 + ) (77, + 77 6 ) = 0, 



which proves what we already knew, namely, that a and 77 are at right angles 

 (S 293). 



We also have 



2/0 (l?3 + ^4) + ^0 (l?5 + %) = 0, 



which proves the following theorem : 



If the instantaneous screw have zero-pitch then the centre of gravity of 

 the body lies in the plane through the instantaneous screw and perpendicular 

 to the impulsive screw. 



337. Calculation of a Pitch Quadric. 



If a, fi, 7 be three instantaneous screws it is required to find with respect 

 to the principal axes through the centre of gravity, the equation to the pitch 

 quadric of the three-system which contains the three impulsive screws corre 

 sponding respectively to a, @, 7 . The co-ordinates of these screws are 

 expressed with reference to the six principal screws of inertia. 



We make the following abbreviations : 



A = a? (a* - 2 2 ) + b* ( 3 2 - 4 2 ) + c 3 ( 5 2 - a 6 2 ), 

 B = a? (&*- /3 2 2 ) + 6 3 (/3 3 - - &*) + c s (& 2 - & a ), 

 C=a? ( 7l 2 - 



