410 THE THEORY OF SCREWS. [372- 



Jf, by imposition of a restraining wrench-chain, the mass-chain continues 

 to twist about the same screw-chain 6, the restraining wrench-chain has 

 neutralized the acceleration. It follows that the restraining wrench-chain, 

 regarded as impulsive, must have generated an instantaneous twist velocity 

 on the accelerating screw-chain, equal and opposite to the acceleration that 

 would otherwise have taken place. The co-ordinates of this impulsive 

 wrench-chain are proportional to 



dT 1 dT 



Pl d0 l &quot; &quot; Pnd0 n &quot; 



The corresponding instantaneous screw-chain is obtained by multiplying 

 these expressions severally by 



and thus we find, as before, for the co-ordinates of the accelerating screw- 

 chain 



373. Accelerating Screw-chain and instantaneous Screw-chain. 



We have, from the expressions already given, 



. dT rlT 



M(utf& + ...+ utfjj = e, JL + . . . e n j* , . 



But the right-hand side is the emanant which we know to be zero, whence 



This shows that 1} ... 6 n , and lf ... d n are on conjugate screw-chains of 

 inertia, and hence we deduce the following theorem : 



Whenever a mass-chain is moving without the action of external forces, 

 other than from constraints restricting the freedom, the instantaneous screw- 

 chain and the accelerating screw-chain are conjugate screw-chains of inertia. 



374. Permanent Screw-chains. 



Reverting to the general system of equations ( 366) we shall now in 

 vestigate the condition that 6 may be a permanent screw-chain. It is obvious 

 that if 0J, ... n are all zero, then 



dT dT_ 

 d0 i &quot; &quot; d0 n 



must each be zero. If, conversely, the differential coefficients just written 

 are all zero, then the quantities j , ... n must each also vanish. 



