690 SCIENCE AND HYPOTHESIS 



of these velocities. This decomposition can only take place in one 

 way. The first of these terms, which I shall call U, will be potential 

 energy ; the second, which I shall call T, will be kinetic energy. It is 

 true that if T+U is constant, so is any function of T+U, <f> (T+U). 

 But this function <j> (T+U) will not be the sum of two terms, the 

 one independent of the velocities, and the other proportional to the 

 square of the velocities. Among the functions which remain constant 

 there is only one which enjoys this property. It is T+U (or a linear 

 function of T+U), it matters not which, since this linear function 

 may always be reduced to T+U by a change of unit and of origin. 

 This, then, is what we call energy. The first term we shall call poten- 

 tial energy, and the second kinetic energy. The definition of the two 

 kinds of energy may therefore be carried through without any am- 

 biguity. 



So it is with the definition of mass. Kinetic energy, or vis viva, 

 is expressed very simply by the aid of the masses, and of the relative 

 velocities of all the material points with reference to one of them. 

 These relative velocities may be observed, and when we have the ex- 

 pression of the kinetic energy as a function of these relative veloci- 

 ties, the co-efficients of this expression will give us the masses. So in 

 this simple case the fundamental ideas can be defined without diffi- 

 culty. But the difficulties reappear in the more complicated cases if 

 the forces, instead of depending solely on the distances, depend also 

 on the velocities. For example, Weber supposes the mutual action of 

 two electric molecules to depend not only on their distance but on their 

 velocity and on their acceleration. If material points attracted each 

 other according to an analogous law, U would depend on the velocity, 

 and it might contain a term proportional to the square of the velocity. 

 How can we detect among such terms those that arise from T or U? 

 and how, therefore, can we distinguish the two parts of the energy? 

 But there is more than this. How can we define energy itself? "We 

 have no reason to take as our definition T + U rather than any 

 other function of T+U, when the property which characterized T+U 

 has disappeared namely, that of being the sum of two terms of a 

 particular form. But that is not all. We must take account, not only 

 of mechanical energy properly so called, but of the other forms of 

 energy heat, chemical energy, electrical energy, etc. The principle 

 of the conservation of energy must be written T+U+Q a con- 

 stant, where T is the sensible kinetic energy, U the potential energy 

 of position, depending only on the position of the bodies, Q the internal 

 molecular energy under the thermal, chemical, or electrical form. 

 This would be all right if the three terms were absolutely distinct; 

 if T were proportional to the square of the velocities, U independent 

 of these velocities and of the state of the bodies, Q independent of the 

 velocities and of the positions of the bodies, and depending only on 



