in the Kinetic Theory of Gases. 97 



after impact has for the same molecules the values 



00^ V-> *; F , V r , ¥, and 00?, y, 0, w-f , v-rf, w—g, t), 



for which we will write 0' and X '. Infinitely small magni- 

 tudes of the order St may be neglected (so that in the two last 

 expressions, for example, t may be written instead of t + St), 

 since the quantities 0, 1; 0', and 0/ occur in equation (18) 

 multiplied by a factor which is itself an infinitely small quan- 

 tity of the order St. By each of these impacts, therefore, 20 

 is diminished by + X ; on the other hand, it is increased by 

 0' + 0x'. By all the selected impacts together, therefore, 20 

 is increased by 



^(0' + 1 '-0-0 1 ). 



We will now consider, together with the action of the hitherto 

 selected impacts, the action of those which in the volume- 

 element do during the time St have acted in exactly the 

 opposite way, and which, for shortness, we will call the re- 

 versed impacts. These latter, therefore, will be just those for 

 which the velocity-components of the centre of gravity and 

 the position of the line of centres of the spheres at the moment 

 of impact lie between the same limits as for the first selected, 

 but for which the velocity-components of the first impinging 

 molecule at the moment at which the impact begins lie between 

 the limits (13); so that those of the second of the impinging 

 molecules differ infinitely little from u— %', v—rf, w—Q. On 

 the other hand, as we at once see, for the reversed impacts 

 the velocity-components of the first of the impinging mole- 

 cules after impact lie between the limits (2) ; but those of the 

 second differ infinitely little from u — £, v—rj, w—%. By each 

 of the reversed impacts, therefore, 20 is increased by 



+ 1 -0'-0 1 '; 



and if we denote the total number of reversed impacts by dn\ 

 then during the time St their effect is to increase 20 by the 

 amount 



<fo'.(0 + 1 -0'-0 1 '). 



This quantity, therefore, by all the first selected impacts and 

 the corresponding reversed impacts, is increased by 



A=(dn-rfn')(0' + 0i'-0-0i). • • • (1*) 

 Exactly as in formula (11) we have next 



dn' '= o-yTi'V cos 6 do da 1 dp.dk St, . . (15) 

 Phil. Mag. S. 5. Vol. 25. No. 153. Feb. 1888. H 



