in the Kinetic Theory of Gases. 99 



all possible values, which we will express by a single integral 

 sign. 



In this, however, we must observe that, after performing 

 the integration, we have counted each impact four times. 

 Because we have extended the integration over all the first 

 selected impacts, we have already counted all the impacts 

 once ; by further extending the integration over all the re- 

 versed impacts, we have counted all the impacts a second 

 time ; moreover, both in the first and in the second integration 

 each separate impact is counted twice, since the molecule with 

 the velocity-components £ r) £ may be both the first as well as 

 the second of the impinging molecules. 



Therefore, on the whole, 



S£cf> = i <r 2 8t £(</>' + <fr' - <f) - &) (//, -/'//) V cos 6 do dco dp dk. (18) 



The whole increase S^$, which X<£ undergoes during the 

 time St, is the sum of the expressions (5), (6), (7), and (18). 

 Since the function </> was left undetermined, there is nothing 

 now to prevent our putting 



where I denotes the natural logarithm. Then we shall have 

 h£$=htW t .dod<o (19) 



This is simply the change which the total number of mole- 

 cules contained in the space suffers in the time Bt ; and is 

 evidently equal to zero, since we assume that the walls of the 

 vessel are solid. 



According to formula (6), S 3 2<£ consists of three summa- 

 tions. Integrating the first of these with reference to x, the 

 second with reference to y, and the third with reference to z, 

 we obtain 



S£<j> = St§fndsda>i (20) 



where ds is a surface-element of the vessel, n the velocity- 

 component of a molecule normal to the surface-element in 

 question. 



We may now for each surface-element ds introduce three 

 new variables instead of £, rj, £, namely the components of the 

 velocity in the direction of normals to the vessel, and in two 

 directions at right angles, which we will denote by n, q, r. 

 Integral (20) then becomes 



St^fndsdndqdr, (20a) 



H 2 



