372] THE THEORY OF PERMANENT SCREWS. 409 



For, if 6-^, ... d,i be the co-ordinates of the displacement, the change in 

 Tis 



^.+ +0 ~ 



which may be written 



O dT- i0 L&amp;lt;W-. 



but this will be zero if, and only if, the screw-chain #/, ... n be reciprocal 

 to the screw-chain 



^ dT ^dT^ 



p l ddj &quot; p n dQn 



371. The Accelerating Screw-chain. 



When the mass-chain has forsaken the instantaneous screw-chain 6, and 

 is twisting about another instantaneous screw-chain &amp;lt;, there must be a 

 twist velocity about some screw-chain p, which, when compounded with the 

 twist velocity about 6, gives the twist velocity about &amp;lt;/&amp;gt;. When &amp;lt; and are 

 indefinitely close, then p is the accelerating screw-chain. 



Taking the n principal screw-chains of inertia as the screws of reference 

 and assuming that external forces are absent, we have 



,i dT 



11 = dl 7 



,if 20 - 

 a u n u n 



It is plain that the co-ordinates of the accelerating screw-chain are 1 , ... n , 

 whence we have the following theorem: 



If a mass-chain be twisting around a screw-chain 6, and if external forces 

 are absent, the co-ordinates of the corresponding accelerating screw-chain are 



proportional to 



I dT I dT 



372. Another Proof. 



It is known from the theory of screw-chains (357) that if a quiescent 

 mass-chain receive an impulsive wrench-chain with co-ordinates 



11 2 11 2 



til U-n 



TT^ 1 &quot; pn&amp;gt; 



P\ Pn 



the mass-chain commences to twist about the screw-chain, of which the 

 co-ordinates are 



pi, Pn- 



