404 THE THEORY OF SCREWS. [363- 



but from the physical property of the kinetic energy already cited, it appears 

 that this kind of displacement cannot change the kinetic energy, whence 



A dT , dT _ 



&quot; n ~ 



364. The Converse Theorem. 



Let us take the general case where the co-ordinates are oc l ,...x n and 

 x l , ... x n . Suppose that cb 2 , ... x n are all zero, then x^ is the velocity of the 

 mass-chain. We shall also take x z , ... x n to be zero, so that we only consider 

 the position of the mass-chain defined by x l . Think now of the two positions 

 for which ^ = and x 1 = x 1 respectively. Whatever be the character of the 

 constraints it must be possible for the mass-chain to pass from the position 

 X-L = to the position x l #/ by a twist about a screw-chain. The magnitude 

 Xi is thus correlated to the position of the mass-chain on a screw-chain about 

 which it twists. 



If the co-ordinates are of such a kind that the identical equation which T 

 must necessarily satisfy has the form 



. dT . dT _ 



Xl dxJ~ Xn dx n ~ 



then for the particular displacement corresponding to the first co-ordinate, 

 # 2 , ... x n are all zero, and 



and as T must involve x^ in the second degree, we have 



T = Hx* 



where H is independent of #/ 



Let #j be the twist velocity about the screw-chain corresponding to the 

 first co-ordinate, then, of course, A being a constant, 



T = Ae*, 

 whence A 6f = Hxf, 



VZtfj = A/774, 

 and by integration and adjustment of units and origins 



& . 



We thus see that while the displacement corresponding to the first co-ordinate 

 must always be a twist about a screw-chain, whatever be the actual nature of 

 the metric element chosen for the co-ordinate, yet that when the identical 

 equation assumes the form 



