Prof. Clausius on the Conduction of Heat by Gases. 427 



various directions of motion, it is easy to see that if the original 

 system of motion were such that an equal number of atoms moved 

 in each direction, this could no longer be the case in the modified 

 system of motion, but that more molecules must move in the 

 directions for which /jl is positive than in those for which /jl is 

 negative. 



In order to be able to express this modification, let us begin 

 by considering the original system of motion, and let us deter- 

 mine the number of molecules whose directions form, with the 

 axis of ,x } angles lying between a and da., the difference between 

 these values being infinitely small. For this purpose let us 

 imagine a spherical surface described with the radius 1 ; let the 

 point, where it is cut by a straight line drawn through the centre 

 in the direction of positive x, be the pole ; and, with the pole for 

 centre, and the arcs a. and ct + dot. for radii, let circles be drawn 

 upon the spherical surface : these two circles will then enclose 

 between them an infinitely narrow zone. The number of mole- 

 cules, whose directions form with the axis of x angles between a 

 and a + da., will then be the same fraction of the entire number 

 of molecules that the area of the surface of the described zone is 

 of the entire area of the spherical surface, and will be represented 

 by 



27rsina& 1 . , 

 -. — , or - sin a da. 



But since ada= — d cos a = — dX, we may also say that the 

 number of molecules whose cosine lies between X and dX is ex- 

 pressed as a fraction of the whole number by 



To find a corresponding expression for the number of mole- 

 cules in the modified system of motion whose cosine lies between 

 /jl and fi — d/j,, we must modify the last expression by the addi- 

 tion of a factor which is dependent upon /jl. Let this factor be 

 H, when the new expression becomes 



iUdfi. 



The factor H may be determined as follows. Since the cosine 

 X is changed into /jl by addition of the component velocity pe, 

 and, similarly, the cosine X + dX into /jl + d/i, the same number 

 which, before the modification, expressed the molecules whose 

 cosine lay between X and X -f dX, will, after the modification, 

 express those whose cosine lies between /jl and /jl + dfi. We may 

 therefore put 



\Hd/i = \ dX, 



whence TT dX 



H =^ ( 6 ) 



