363] 



THE THEORY OF PERMANENT SCREWS. 



403 



Then we obtain, by elimination of Bx l} ... 8x n , and 8e, 



AT, 



AA, 



dT dT 



df n dfn 



= 0. 



dx l &quot; dx 1t 



Such is the general condition which must be satisfied by the kinetic energy 

 of any material arrangement whatever. But the equation is so complicated 

 when expressed in ordinary rectangular co-ordinates that there is but little 

 inducement to discuss it. 



363. The Identical Equation in Screw-chain Co-ordinates. 



The Theory of Screw-chains exhibits this equation in a form of special 

 simplicity. For, suppose that 



then the equation of the last article reduces to 



We thus have the following theorem : 



If the co-ordinates, #/ , ... O n , of a mass-chain be n twists about n screiv- 

 chains, belonging to the system of screw-chains which express the freedom of 

 the mass-chain, and if Oi, ... O n be the twist velocities of the mass-chain about 

 these same screw-chains, then the kinetic energy T satisfies the equation 



A dT 



v i ~jm~&amp;gt; f Un ja~ f ~ 

 a PJ au n 



I have thought it instructive to exhibit the origin of this equation as 

 a special deduction from the case of co-ordinates of the general type. For 

 a brief demonstration the following simple argument suffices : 



If the mass-chain be displaced through S0/, ... S0 n while the velocities 

 are unaltered, the change of kinetic energy is 



If the change of the position be due to a small twist Se around the screw- 

 chain with co-ordinates l} ... 6 n , then 



262 



