240 Prof. Ball, Note on Geomett'ical Mechanics. [Oct. 29, 



The following Communications were made to the Society : 



(1) Note on Geometrical Mechanics. By Prof. Sir Robert 

 S. Ball. 



The kinetic energy of a material system must always submit 

 to a certain identical equation. Take as a particular case a body 

 rotating around a fixed point. Let x^, x^, x^ be the coordinates 

 (supposed small) expressing the position of the body as defined 

 by rotations about three intersecting axes and x^, x^, x^ the angular 

 velocities. If the body before being set to spin had been dis- 

 placed by a small rotation about the instantaneous axis and then 

 set to spin with the original angular velocity, the effect on the 

 kinetic energy will be nil. For the displacement merely altering 

 the azimuth of the body about the axis cannot affect the energy. 

 This alteration however does affect the coordinates x^, x^, x^ by 

 changing them into ^, + ei\, x,^ + €x^, x^+ ex^ when e is small. 

 Introducing these values into T we have 



. dT . dT ., dT _ 



1 ZJ^ "^ 2 Tj ^ 3 i7 — ' 



ka/OC. (X/OGf. iAjOC„ 



more generally it can be shown that in any material system what- 

 ever 



. dT . dT^_ 



X^ 7 + . . . + X^^ -J \Jy 



when screw coordinates are employed. In ordinary coordinates an 

 equivalent equation holds, but its form is complicated. Whenever 

 the equation admits of the above simple form the coordinates 

 must be screw coordinates. The function is an evectant retaining 

 its form for any linear transformation inasmuch as x^, x,^, &c. x^^ 

 axe cogredient with x^, x^, &c. x^^. See T.R.I. A. Vol. xxix. p. 613. 



Here we give a dynamical illustration. Let the case taken 

 be a rigid body with three degrees of freedom, and let the screws 

 of reference be coreciprocal. Then we have 



. dT . dT . dT _ 



* dx^ ^ dx^ ^ dx^ 



The wrench with coordinates 



Vi ' dx^ ' P^ ' dx^ ' Pz ' dx^ ' 



is such that if applied to the body it will prevent the body from 

 forsaking the instantaneous screw about which it is twisting. It 

 acts so to speak as a restrainer. 



