CRITICAL NOTE. 143 



to every body of ordinary dimensions the property of mass. Whether any 

 physical meaning can be attached to the phrase "mass of a molecule" is not 

 a question with which we need here concern ourselves. 



In the next place we have arrived at certain quantities which we call quanti- 

 ties of energy, and we have admitted the existence of two kinds of energy, one 

 depending on motion, the other on position. In our theory the notion of energy 

 was derived by a long process from the notion of force. Now, if we had deve- 

 loped our theory a little further by purely analytical processes, we should have 

 found, at least for conservative systems, that the equations of motion of such 

 a system could be deduced by a particular process of differentiation from a 

 knowledge of two analytical expressions, one representing the kinetic energy, 

 and the other the potential energy. The method by which this can be done 

 is, in fact, the method of the variational equation indicated in Article 159. 

 The virtual work of the forces, which was the right-hand member of the 

 equation, was shown to be derivable by differentiation from the expression 

 for the potential energy. The virtual moment of the kinetic reactions 

 can be shown to be derivable by certain differentiations from the kinetic 

 energy. This important result is due to Lagrange. It thus appears that 

 from the expressions for the kinetic and potential energies of a conser- 

 vative system alone, if properly set down, all the equations of motion can be 

 deduced. In particular the equations of motion of the bodies of the solar 

 system can be so deduced. It is absolutely necessary to attribute to every 

 body of ordinary dimensions the possession, in general, of kinetic energy, and 

 to attribute to every system of bodies the possession of energy, only a part of 

 which is kinetic. 



Here then we seem to have some grounds for the kind of compromise we 

 are seeking. The suggestion is that we should retain the conceptions of mass 

 and energy, and the separation of forms of energy, one form being kinetic. 

 In accordance with this suggestion mass will be a property of bodies which 

 affects their motions precisely as before ; to formulate this property we have 

 merely to make the mass of a body a constant coefficient of a term in the 

 kinetic energy ; further force will be an expression for the amount of mutual 

 action, which will be derivable from the expression for the potential energy 

 whenever the system is conservative ; and thus, in many cases, observed facts, 

 stated in the beginnings of the science in terms of resultant forces, can be 

 stated in the same way still ; and we have seen that, except for gravitation, 

 nothing about forces other than resultants can be stated both with precision 

 and so as to agree with facts of observation. 



Now if we were asked what this mass or energy that we are talking 

 about is, we should have to confess that we did not precisely know, that we 

 could not expect to know until we had explored and formulated the whole 

 domain of physical science. We should have to say that the notions were 

 reached by means of a provisional theory in which they had perfectly definite 

 meanings; that they had been further generalised because the old theory 

 succeeded in representing some facts, and failed in representing others, and 

 in such representations these two were the most important among the really 



