2 CLAUSIUS ON THE MECHANICAL EQUIVALENT 



these have been most frequently employed and upon the largest 

 scale. In such an investigation, however, it appears to me 

 more suitable to commence with machine electricity ; although 

 this subject presents greater difficulties to mathematical treat- 

 merit, still in principle it is the most simple ; for here we have 

 to do with electricity alone, unaccompanied by the incidental 

 actions of chemistry and magnetism. 



In the following pages I have endeavoured to reduce the 

 effect produced by an electric discharge to a definite unit derived 

 from the principles of mechanics, and have compared in certain 

 simple cases the result arrived at with those of experiment. 

 The coincidence has, as will be observed, been so satisfactory, 

 that in my opinion the deduced result is not only an undoubted 

 law of electric action, but also a new corroboration of the me- 

 chanical theory of heat. 



Let us imagine a system of material points possessing the 

 masses m, m', W, &c., and having, in a rectangular system of 

 coordinates, at a certain point of time /, the coordinates x,y, Z', 

 a^, ?/', z^ ; a?", 2/", z", &c. Let these masses be acted upon by a 

 system of given forces, and let the components of the total force 

 which acts upon m be X, Y, Z, and for m', X', Y', Z', &c. Let 

 the points be either quite free to move, or else limited in their 

 motions, which last will of course be the case when the points 

 are in any way connected together, as also when certain exte- 

 rior conditions of motion exist ; for example, if one of the points 

 should be compelled to move in a certain surface, or along a cer- 

 tain line. The conditions, however, must not be such, that by 

 them alone, and without the exercise of the given forces, motion 

 can take place ; w hich, for instance, would be the case if the 

 surface or line just supposed, and which the point cannot forsake, 

 were itself in a state of motion, dr such that the motion of the 

 given masses should be able to set other masses in motion which 

 are not embraced in the system. That is, in other words, the 

 moving forces and the masses moved by them must be given expli- 

 citly. Finally, let the velocities of the masses m, m', m", &c. at 

 the time t, be denoted by v, v\ ?;", &e., we then obtain the fol- 

 lowing general expression, 



^^md{f)z:zl,{:^dx^Xdy^r'Ldz), . . . . (1) 



