38 TH DYNAMICAL THEORY OF OASES. 



The Telocity of any of the dN t molecules of the first system relative to 

 the dtf t molecules of the second system is V, and each molecule J/, will in 

 the time & describe a relative path F& among the molecules of the second 

 system. Conceive a space bounded by the following surfaces. Let two cylindrical 

 Mrfew* have the common axis F& and radii b and b + db. Let two planes 

 be drawn through the extremities of the line V8t perpendicular to it. Finally, 

 let two planes be drawn through VSt making angles tf> and </> + (//> with a plane 

 through V parallel to the axis of x. Then the volume included between the 

 four planes and the two cylindric surfaces will be Vbdbd<fit. 



If this volume includes one of the molecules M t , then during the time 8* 

 there will be an encounter between M l and J/,, in which b is between b and 

 h + db, and <f> between <f> and <f> + d<f>. 



Since there are <&V, molecules similar to 3f, and cZiV, similar to M t in unit 

 of volume, the whole number of encounters of the given kind between the two 

 systems will be 



Now let Q be any property of the molecule M lt such as its velocity in a 

 given direction, the square or cube of that velocity or any other property of the 

 molecule which is altered in a known manner by an encounter of the given 

 kind, so that Q becomes Qf after the encounter, then during the time 8< a 

 certain number of the molecules of the first kind have Q changed to Q', while 

 the remainder retain the original value of Q, so that 



8QdN, = (<?- 



or -(V-tyVtotod+UfjUf* ..................... (3). 



Here - * ' refers to the alteration in the sum of the values of Q for the 

 , molecules, due to their encounters of the given kind with the dN, molecules 



of the second sort. In order to determine the value of ^ ', the rate of 



ot 



alteration of Q among all the molecules of the first kind, we must perform 

 the following integrations : 



1st, with respect to <f> from <j> = to ^ = 2rr. 



2nd, with respect to b from 6 = to 6 = oo . These operations will give 



