362] THE THEORY OF PERMANENT SCREWS. 401 



362. A Property of the Kinetic Energy of a System. 



It is obvious that the mere alteration of the azimuth about a fixed axis 

 from which a rigid body is set into rotation will not affect its kinetic energy, 

 provided the position of the axis and the angular velocity both remain un 

 altered. 



A moment s reflection will show that this principle may be extended to 

 any movement whatever of a rigid body. At each instant the body is 

 twisting about some instantaneous screw a with a twist velocity a. Let 

 the body be stopped in a position which we call A. Let it receive a dis 

 placement by a twist of any amplitude about a and thus be brought to a 

 position which we call B*. Finally, let the body be started from its new 

 position B so as to twist again about a with the original twist velocity d, 

 then it is plain that the kinetic energy of the body just before being stopped 

 at the position A is the same as its kinetic energy just after it is started 

 from the position B. 



Enunciated in a still more general form the same principle is as 

 follows : 



Any mass-chain in movement is necessarily twisting about some screw- 

 chain. If we arrest the movement, displace the mass-chain to an adjacent 

 position on the same screw-chain, and then start the mass-chain to twist 

 again on the same screw-chain, with its original twist velocity, the kinetic 

 energy must remain the same as it was before the interruption. 



This principle requires that whatever be the symbols employed, the 

 function T, which denotes the kinetic energy, must satisfy a certain identical 

 equation. I propose to investigate this equation, and its character will 

 perhaps be best understood by first discussing the question with co-ordinates 

 of a perfectly general type. We shall suppose the mass-chain has n degrees 

 of freedom. 



Let the co-ordinates x lt ...x n represent the position of the mass-chain, 

 and let its instantaneous motion be indicated by x l ,...x n . Let be the 

 initial position of the mass-chain, then in the time Bt it has reached the 

 position , whereof the co-ordinates are 



x l + x^t, ... x n + x n $t. 



The movement from to must, like every possible movement of a 

 system, consist of a twist about a screw-chain. This is a kinematical fact, 

 wholly apart from whatever particular system of co-ordinates may have 



* We have supposed that the pitch of this displacement is the same as the pitch of a. This 

 restriction is only introduced here because the constraints will generally forbid the body to make 

 any other twist about the axis of a. If the body were quite free we might discard the restriction 

 altogether as is in fact done later on ( 376). 



B. 26 



