374 MATHEMATICAL APPENDICES 11* 



only the former, the principle of the conservation of energy, 

 requires a few remarks to be made, since it may be applied in 

 different ways according as the molecules are to be looked on as 

 simple massive points or as made up of atoms. In the former 

 case, of the frwo_ kinds of energy, kinetic and potential, the sum 

 of which, according~tcTthe principle, possesses a value TiEat is 

 constant for all time and under all circumstances, only the former 

 comes into account ; for since two molecules act on each other 

 only at the moment of a collision, the potential energy can be 

 neglected if we take into account in the calculation those speeds 

 with which the particles move betiveen two collisions, and not 

 ~X during a collision. W T e have then to consider simply the sum of 

 the kinetic energies as invariable, or to introduce the theorem that 

 no kinetic energy is lost at a collision. 



We shall first of all limit our consideration to this simpler case, 

 and postpone that of composite molecules for later investigation 

 in 21*. 



12*. Determining Equations 



By 10* our problem consists in finding the function 

 F(u, v, w) which has the property that the product 



F(u { , v } , w^F(u^ v 2 , iu 2 ) . . . F(u N , V N , W N ) 



has the same constant value C for all values of the components 

 u, v, w of the molecular velocities that occur together. According 

 to the last discussion the values of these components are not 

 magnitudes that vary arbitrarily and independently, but they are 

 subject to the conditions that they must satisfy the two named 

 theorems of mechanics. If, then, we denote by E the mean value 

 of the kinetic energy of one of the N molecules in question, by m 

 the mass of a molecule, and finally by a, b, c the components of 

 the velocity with which the centroid of the whole system of 

 gaseous particles moves, the two theorems are expressed by the 

 equations 



Na= u { + u 2 + . . . + U N ] 



Nc = w { + w. 2 + . . . + W N ) 

 For our problem these equations represent the conditions con- 



