XII. — On the Definition of Work Done. — By Prof. J. G. 



MacGregor, Dalhousie College, Halifax, N. S. 



(Received 1st May 1895.) 



The usual definition of work done is the product of the 

 magnitude of a force into the component in its action line of 

 the displacement of its point of application. Positive work is 

 said to be done by the force on the body on which it acts, if the 

 force and the competent displacement have the same direction ; 

 negative work, if they have opposite directions. Positive work 

 is said to be done hy the body against the force if the force and 

 the component displacement have opposite directions ; negative 

 work, if they have the same direction. Thus work done by the 

 body is just work done by the force with the sign changed. 



Prof. Simon Newcomb has pointed out * that as no axes of 

 reference are specified in the above definition, the displacement 

 referred to in it, and therefore the work done, as determined by 

 it, are quite arbitrary. He has therefore proposed to give the 

 definition the following form : — 



" The work done by a force is the product of the intensity of 

 the force into the amount by which the two material points 

 between which it acts approach to or recede from each other ; 

 the work being positive when the approach or recession is in 

 the direction of the force, negative in the opposite case." 



It has been held by many writers "f* that the Laws of Motion 

 require, for their complete enunciation, the specification of the 

 axes of co-ordinates, by reference to which they hold, any such 

 system of axes being called a dynamical reference system. If 

 in the definition of work done also, the axes of reference Avhich 

 are to be employed, be specified, its arbitrary character will 

 disappear ; for by reference to given axes, any displacement will 



*Philosophical Magazine, Ser. 5, Vol. xxvii (1889), p. 115. 

 tSee Phil. Mag, Ser. 5, Vol. xxxvi (1893), p. 233. 



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