81] THE PRINCIPAL SCREWS OF INERTIA. 7i 



80. Impulsive Screws and Instantaneous Screws. 



If a free quiescent rigid body receive an impulsive wrench on a screw 77, 

 the body will immediately commence to twist about an instantaneous screw 

 a. The co-ordinates of a being given for the six screws of reference just 

 denned, we now seek the coordinates of 77. 



The impulsive wrench on 77 of intensity 77 &quot; is to be decomposed into com 

 ponents of intensities tj &quot;r} 1 , ... i} &quot;rj 6 on ( l , ... w s . The component on co n 

 will generate a twist velocity about o&amp;gt; H amounting to 



jL W 



M Pn 



but if a be the twist velocity about a which is finally produced, the expression 

 just written must be equal to aa n , and hence we have the following useful 

 result : 



If the co-ordinates of the instantaneous screw be proportional to a l} ... a fi&amp;gt; 

 then the co-ordinates of the corresponding impulsive screw are proportional to 



81. Conjugate Screws of Inertia. 



Let a be the instantaneous screw about which a quiescent body either 

 free or constrained in any way will commence to twist in consequence of 

 receiving an impulsive wrench on any screw whatever 77. Let ft be the 

 instantaneous screw in like manner related to another impulsive screw 



We have to prove that if be reciprocal to a then shall 77 be reciprocal 

 to j3. 



When the body receives an impulsive wrench on of intensity &quot;&quot; there 

 is generally a simultaneous reaction of the constraints, which takes the form 

 of an impulsive wrench of intensity ///&quot; on a screw /*. The effect on the body 

 is therefore the same as if the body had been free, but had received an 

 impulsive wrench of which the component wrench on the first screw of 

 reference had the intensity ^ &quot;^ + p, &quot;^ . This and the similar quantities 

 will be proportional to the co-ordinates of the impulsive screw which had the 

 body been perfectly free would have /3 as an instantaneous screw. These 

 latter, as we have shown in 80, are proportional to p l /3 1 ,p.,@. i ...p 6 /3 6 . Hence 

 it follows that, h being some quantity differing from zero, we have 



Multiplying the first of these equations by p^, the second by^oa^, &c. adding 



