in the Kinetic Theory of Gases. 95 



formed of the seven variables x, y, z, %, rj, £. t, which we will 

 denote by <j). This function may be chosen quite arbitrarily. 

 It may be altogether independent of the function /, or may be 

 connected with it in any way whatever (e. g. it may be iden- 

 tical with /, or equal to the natural logarithm of/, &c.) . If 

 we substitute in <£ the coordinates and component velocities 

 of the centre of any molecule, we obtain the value of cf> corre- 

 sponding to the molecule in question at the time t. The sum 

 of the values of </> corresponding to all the molecules at the 

 time t we will denote by X (</>). 

 We may then evidently write 



£<£=]>./. do dco, (4) 



where the sign \ denotes an integration with reference to the 

 variables £, rj, £ from -co to +<x> , but with reference to the 

 variables x, y, z over the whole volume of the vessel. A 

 change of the sum 2</> will be produced during an infinitely 

 small time 8t by various causes : — 



(1) By the function <£ explicitly containing the time. The 

 change thus produced may be denoted by a prefixed 8 V "We 

 have then 



8 1 t<j> = 8t^.fdod(o (5) 



(2) Because the molecule which at the time t has the co- 

 ordinates x, y, z and the velocity-components %, tj, £, at the 

 time t + 8t has the coordinates x + %8t, y + rjSt, z + $8t and the 

 velocity-components f + XS£, r] + Y8t, £+Z&. The whole 

 change in S<£ thus produced will be 



where 



s^=s«J( f || + ,g+r||)*^ . (6) 



^-^(Xg + Yg + Zlf)*,*.. . (7) 



If we employ the abbreviation 



(B+«8+'g + # +x S +T S + *g)»- , *ffl> 



then 



8£<j> + 8£(f> + 8 z Z<j: = §8'(f>fdodGy. . . . (9) 



