430 Prof. Clausius on the Conduction of Heat by Gases, 



ties are also somewhat unequal. We may hereafter denote the 

 arithmetical mean of a magnitude whose value, in the particular 

 cases which occur, is various, by making a horizontal stroke over 

 the symbol which represents the particular values of the magni- 

 tude, so that V shall represent the mean value of V, and s and s 2 

 the mean values of s and s 2 . We may then write 



— dV ld 2 U 2 -5 , m 



V = U -^ S+ 2^^ S - ( 10 > 



In this expression it is to be observed that the magnitude s 2 

 is not equivalent to (s) 2 , but that it must be specially deter- 

 mined. Thence it also follows that the mean values of the 

 powers V 2 , V 3 , &c. are not quite equal to the corresponding 

 powers of the mean value V. We must, in fact, in order to ob- 

 tain this mean value, start from the equation (9), and, after 

 having squared it, cubed it, &c, then put the mean values for s, 

 s 2 , &c. We thus obtain 



V 4 =&c. 



The differences between the magnitudes V 2 , V 3 , &c, and the 

 magnitudes (V) 2 , (V) 3 , &c, which latter are obtained by squaring, 

 cubing, &c, equation (10), occur, as will be seen, first in 

 those terms which are of the second degree in relation to the 

 length of the excursion s ; and as these excursions are, on the 

 average, very small quantities, the differences are also very small. 



§ 9. It now becomes necessary to determine the values of s 

 and s 2 with greater exactness. 



To this end, we will first examine the behaviour of these mag- 

 nitudes when the temperature and density of the given quantity 

 of gas are uniform throughout, and will afterwards superadd the 

 modification due to the inequality of temperature and density. 



Considering, then, all the molecules which are contained at 

 any given time in a stratum of a gas whose temperature and 

 density are everywhere the same, we ask ourselves, how great 

 are the distances which the several molecules have traversed be- 

 tween their last impact and the moment at which we consider 

 them. The likelihood that a molecule has traversed a distance 

 lying "between s and s + ds, between its last impact and the 

 moment fixed upon, is just as great as the likelihood of its tra- 

 versing an equal distance between this moment and its next 



