366] THE THEORY OF PERMANENT SCREWS. 405 



the metric element must be essentially the amplitude of the twist about the 

 screw-chain. We have thus proved the following theorem : 



The co-ordinates must be twists about n screw-chains of reference whenever 

 the identical equation, satisfied by T, assumes the form 



6 dT . a dT_ 

 9l &quot; ffn ~ 



365. Transformation of the Vanishing Emanant. 



Suppose that the position and movement of a mass-chain were represented 

 by the co-ordinates #/, #&amp;lt;/, n ; #i&amp;gt; @2, - &n when referred to one set of 

 n screws of reference, and by &amp;lt;/&amp;gt;/, (f&amp;gt; 2 } ... &amp;lt;,/; &amp;lt;/&amp;gt;!, &amp;lt; 2 , &amp;lt;/&amp;gt; when referred to 

 another set of screws of reference. Then of course these sets of co-ordinates 

 must be linearly connected. 



We may write 



4 - (11) ft .-.. + (!)*, , 



= (nl) &amp;lt;/... + (&amp;gt;,)&amp;lt;,;. 



Then, by differentiation 



4 -(ii) ^... 



Thus the two sets of variables are co-gredients, and by the theory of 

 linear transformations we must have 



dT , dT dT : dT 



The expression on either side of the equation is of course known in algebra 

 as an emanant ( 261). 



We could have foreseen this result from the fact that whatever set of 

 n independent screw-chains belonging to the system was chosen, the identical 

 equation must in each case assume the standard form. 



366. The General Equations of Motion with Screw- chain 

 Co-ordinates. 



The screw-chain co-ordinates of a mass-chain with n degrees of freedom 

 are #/, ... d n ; the co-ordinates of the velocities are 1} ... 0*. Let 77 be the 

 wrench-chain which acts on the system. Let the components of the wrench- 

 chain, when resolved on the screw-chains of reference, have for intensities 

 Vi &amp;gt; W- Let p l , ... p n be the pitches of the chains of reference, by which 

 is meant that 2p t is the work done on that screw-chain by a twist of unit 

 amplitude against a wrench of unit intensity on the same screw-chain. Then 

 the screw-chains of reference being supposed to be co-reciprocal, we have, 



