Prof. Clausius on the Conduction of Heat by Gases. 431 



impact; and the likelihood of the latter event can be easily- 

 expressed. 



If, from a given moment of time, a large number of molecules 

 be supposed to move through the gas with an equal velocity, their 

 motion will cause each of them sooner or later to impinge upon 

 other molecules ; and if z denote the number of molecules which 

 traverse the distance s without striking against other molecules, 

 z must diminish according to a definite ratio as s increases. If 

 we say that the probability of one molecule striking another 

 while traversing the infinitely small distance ds is ads, then of 

 the number z which have traversed the distance s without impe- 

 diment, the number zads will be taken up during the next por- 

 tion of their course ds, and the decrement of z will hence be 

 represented by the equation 



dz=—zadsi 



whence it follows that, putting Z for the initial value of z when 

 s = 0, 



z^Ze-™. 



This value being substituted for z in the product zads, gives the 

 expression 



Ze~ as ads 



for the number of molecules the length of whose excursions lies 

 between s and s + ds. 



In order now to obtain the mean length of all the excursions, 

 it is only needful to multiply the last expression by s, then to 

 integrate from s = to s= go, and to divide the integral by the 

 whole number Z. This gives 



se- a 'ads= — (12) 



r 



a 



This expression applies primarily to the mean length of the 

 distances moved through by the molecules between the point of 

 time in question and their next impact ; but it can also be directly 

 used for the distances the molecules have moved through between 

 their last previous impact and the instant in question, for the 

 distances before any given point of time must, on the average, 

 be equal to the distances after it. 



The same value, -, is also obtained if we investigate the mean 



distances traversed between every two impacts during a given 

 time. For if, instead of considering the motions of all the mole- 

 cules between a given instant and their next impacts, we take a 

 large number of impacts as our starting-point, and then follow 

 the motions of the molecules until their next impacts, all the 



