AND OF THE SECOND LAW 49 



co-ordinates concurrently lie between , 77, , and 

 will be represented by the product 



where the only thing known about function / is that the sum of 

 the fractions f()d extended over all the values of must = unity, 

 so that 



Now if we suppose all of the velocities of the n molecules to 

 be laid off as vectors from a pole O, the three directions , y, 

 will constitute about O a perfectly arbitrary system of co-ordi- 

 nates in which (d) (drf) (dt)=dV designates a volume element l 

 and the velocity p of a molecule is given by 



1 In MAXWELL'S distribution the molecules are assumed to be uniformly scat- 

 terred throughout the unit volume; it is the velocities only that are variously 

 distributed in the different elementary regions. To realize the haphazard char- 

 acter (necessary in Calculus of Probabilities) of the motions of the molecules, we 

 must bear in mind that each of the molecules in the unit volume has a different 

 velocity and direction; here no direction has preference over another, i.e., one direc- 

 tion of a molecule is as likely as another. Here at first we write expression for 

 the number of molecules whose velocities parallel to the co-ordinate axes are 

 respectively confined between the velocity limits: 



and 



and y-\-dr), 

 and 



To find the number of molecules thus limited the procedure given above isessentially 

 as follows: Expressed as a fraction f()d = probability of velocities parallel to 

 axis having values between and -fd and expressed as a number nf()d= 

 number of molecules having such velocities between the assigned limits; similarly, 

 probability of velocities parallel to y axis having velocities between y and 

 As these are two independent sets of velocities, the probability of their 

 concurrence is the product f()d-f(r})df) and the number of molecules thus con- 



