Prof. Clausius on the Conduction of Heat by Gases. 519 



tions as being drawn from its centre, so that every point on the 

 surface of the sphere represents a direction. If the molecules 

 moved equally in all directions, the number whose directions 

 would fall within an element dco of the spherical surface would 

 have the same ratio to the whole number of molecules as the 

 size of that element to the surface of the entire sphere ; hence it- 

 would be represented, as a fraction of the whole number of 



molecules present, by — . In the present Case, in which the 



molecules do not move equally in all directions, this expression 

 must undergo a modification, and one of such a kind that, 

 according to the notation adopted in § 12, the number of mole- 

 cules whose directions fall within the superficial element dco will 

 be represented, as a fraction of the entire number of molecules, 

 , ~dco 



7 F , t 



If R represents the mean velocity of the given molecule rela- 

 tively to those molecules whose directions fall within the element 



dco, and R its mean velocity relatively to all the molecules pre-, 

 sent, the following equation will serve to determine the latter 

 quantity : — 



J dco 



R=j^IR. ...... (33) 



The integration must here be extended to the whole spherical 

 surface ; and this integral we will now proceed to develope. 

 § 19. According to equation (29), 



R= \/Y 2 + v*-2Vvcoscp, 

 to which we will give the following slightly modified form : 



• ' " ' R=V2^Vv^l-cos(/) + ( -^^ 2 . . . (34) 



The velocity V of any molecule existing in the stratum differs, 

 as we have already seen, only so slightly from the velocity u of 

 the molecules which move perpendicularly to the axis of x, that 

 the difference is a magnitude of the same order as e. If we now 

 assume that the velocity of the given molecule denoted by v like- 

 wise only differs from u by a magnitude of the same order, the 

 difference V— v must also be a magnitude of the order of e; and 



" (V—vf 

 hence the term n ~ T , which occurs in the last root, must be 



a magnitude of the order of e 2 . By integrating this term no- 

 thing but another term of the same order can be obtained : 



