408 THE THEORY OF SCREWS. [368- 



whcnce we deduce the following theorem : 



If the position of a mass-chain be referred to co-reciprocal screw-chains 

 of reference, then 



1 dT 1 dT 



Pi dffi p n d6n 



are the co-ordinates of the restraining wrench- chain which would coerce the 

 mass-chain into continuing to twist about the same screw-chain 6*. 



369. Physical meaning of the Vanishing Emanant. 



We may verify this theorem by the following method of viewing the 

 subject. It must be possible to coerce the system to twist about 6 by the 

 imposition of special constraints. The reactions of these constraints will 

 constitute, in fact, the restraining wrench-chain. It is, however, a character 

 istic feature that, as the system is, ex hypothesi, still at liberty to twist 

 about 6, the reaction of any constraints which are consistent with this 

 freedom must lie on a screw-chain reciprocal to 0. 



The condition that two screw-chains, 6 and rj, shall be reciprocal ( 354) 



s 



but this is clearly satisfied if for i) 1} ... we substitute 



l^dT^ 1 dT 



piW p n d0 n ] 



for the equation then becomes 



dT i. fl dT n 



i ddr&quot; n de~r 



which, when multiplied by 0, reduces to the known identity 



e dT ^ dT o 



*ap&quot;- + *n?--?.- 



We thus obtain a physical meaning of this equation. It is no more than an 

 expression of the fact that the restraining wrench-chain must be reciprocal 

 to the instantaneous screw-chain. 



370. A displacement without change of Energy. 



It should also be noticed that provided the twist velocities remain un 

 altered the kinetic energy will be unchanged by any small displacement 

 of the mass-chain arising from a twist on any screw-chain reciprocal to the 

 restraining screw-chain. 



* A particular case of this, or what is equivalent thereto, is given in Williamson and Tarleton s 

 Dynamics, 2nd ed., p. 432. 



