298 THE THEORY OF SCREWS. [280- 



Unless in this exceptional case where p^ is infinite it is always true that 

 when p a is zero, a and 77 are at right angles. 



It is universally true that when the impulsive screw and the instan 

 taneous screw are at right angles (the body being quite free), the pitch of 

 the instantaneous screw must be zero. 



For if p a were not zero when cos (a?;) was zero then a. must be zero. As 

 some motion must result from the impulse (the mass of the body being 

 finite) we must have p a infinite. The initial motion is thus a translation. 

 Therefore the impulse must have been merely a force through the centre 

 of gravity ; a and 77 must be parallel and cos (a?;) could not be zero. 



The expression for the kinetic energy in 279, 



cos 



assumes an indeterminate form when the impulsive wrench reduces to a 

 couple. For we then have p a = 0, but as cos (a?;) is not zero the expression 

 for v? ari , i.e. 



2 {(POL + pj cos (CM?) - d ar , sin (arj)}, 

 becomes infinite. 



The expression for the kinetic energy arising from an impulsive wrench 

 of unit intensity on a screw 77 applied to a free body of unit mass which 

 thereupon begins to twist with an instantaneous movement about a screw a 

 has the concise form 



Po 



281. Conditions to be fulfilled by two pairs of Impulsive and 

 Instantaneous Screws. 



Let a be a screw about which a free rigid body is made to twist in 

 consequence of an impulsive wrench administered on some other screw, 77. 

 Let /3 be another instantaneous screw corresponding in like manner to as 

 an impulsive screw. Then we have to prove that the two following formulas 

 are satisfied * : 



cos^) cos W + ~To^ cos () = 2l 



Pa 



/ \ w DT) / / i &amp;lt;- \ ** 



cos (ttq) cos (pt) 



* Proceedings of the Camb. Phil. Soc., Vol. ix. Part iii. p. 193. 



