300 THE THEORY OF SCREWS. [282, 



The formula becomes, when p a is very large, 



Pa- P? 1 / *-\ 



*-r r ura = - ~~ir. I p a cos (a|). 

 cos(cc?7) cos(/3|) &quot; 



In this case as the twist about a is merely a translation, we must have 

 cos err) 1, so that 



is to vanish, but this cannot be secured by making p? zero, because that 

 cannot happen without cos (/3) being zero, except the pitch of be infinite 

 ( 280) which is the case already excluded. It is therefore necessary that 

 cos (a) be zero, but this requires that a and be reciprocal, i.e. that vr a = 0. 



Let us now suppose that we try to satisfy the original equations by 

 making OT^ = 0, pp = 0. Here again we find that Pfs = entails cos (/3) zero, 

 except p$ = GO . This in general makes -CT^ infinite so that the equation is 

 not satisfied. If a and were at right angles then no doubt the equation 

 would be satisfied, but then -S7 a | is zero. We thus see that notwithstanding 

 the special form of the fundamental equation ( 281) it implies no departure 

 from the complete generality of the principle that whenever is^ is zero 

 then must &amp;lt;r f be also zero. 



283. A Fundamental Theorem. 



Let us suppose that a rigid body is either entirely free or constrained in 

 any manner whatever. Let r) be an impulsive screw whose pitch p^ is not 

 infinite. Let 77 &quot; be the intensity of an impulsive wrench on that screw, it 

 being understood that ?/&quot; is to be neither zero nor infinity. Let a be the 

 instantaneous screw about which the body, having been previously quiescent, 

 will commence to twist with an instantaneous twist velocity a. It is also 

 supposed that p a is neither zero nor infinity. 



Let be the impulsive screw similarly related to /9, and let the affiliated 

 symbols have the corresponding significations and limitations. 



Let be the impulsive screw similarly related to 7, and let the affiliated 

 symbols have the corresponding significations and limitations. 



The instantaneous movement of the body must necessarily be the same 

 as if it had been quite free and had received in addition to the impulsive 

 wrench of intensity 77 &quot; on the screw rj, an impulsive wrench of intensity p&quot; 

 situated on some screw p belonging to the system of screws reciprocal to the 

 freedom of the body. 



Let these two wrenches compound into a single wrench of intensity to &quot; 

 on a screw w. 



