86 Prof. Clausius on Molecular Motion, 



the probability of its passing through the latter one. Similarly, 

 for a layer of the thickness 3, we have a^, &c., and for a layer of 

 any thickness a: we may accordingly write a^. Let us transform 

 this expression by putting e~* for a, in which eis the base of the 

 natural logarithms, and — a= log^ a, which logarithm must be 

 negative, because a is less than 1. If now we denote the pro- 

 bability of the free passage through a layer of the thickness x 

 by W, we have the equation 



W = e-% (1) 



and we have only to determine here the constant a. 



Again, let us consider a layer of such thinness that the higher 

 powers of the thickness may be neglected in comparison with 

 the first. Calling this thickness 8 and the corresponding pro- 

 bability Ws, the former equation becomes 



Wj=e-«« = l-«8 (2) 



The probability in this case may also be determined from 

 special considerations. Let us direct our attention to any plane 

 in the layer parallel to one of the bounding planes of the layer, 

 and let us suppose all the molecules whose centres lie in the 

 layer to be so moved perpendicular to the layer that their centres 

 all fall upon this plane ; we have now only to inquire how great 

 the probability is that the point, in its passage through this 

 plane, meets with no sphere of action ; such probability may be 

 simply represented by the proportion of two superficial areas. 

 Of the entire part of the plane which falls within the given space, 

 a certain portion is covered by the gi'eat circles of the spheres of 

 action whose centres fall upon it, while the remaining portion is 

 free for the passage ; and the probability of the uninterrupted 

 passage is therefore expressed by the relation of the free portion 

 of the plane to the whole plane. 



From the manner in M'hich the density was determined at the 

 beginning of this article, it follows that in a layer of thickness 

 \, so many molecules must be contained, that, if they be supposed 

 brought into one and the same plane parallel to the bounding 

 plane, and to be arranged still quadratically in this plane, then 

 the side of the small square in whose corners would be situated 

 the centres of the molecules would be equal to X. Hence it 

 follows, that the part of the plane which would be covered by the 

 great circles of the spheres of action, would be related to the 

 remainder of the plane as a great circle would to a square of 

 side X, so that, accordingly, the covered superficial area would be 

 expressed by the fraction 



of the entire superficial area. In order to ascertain the corre- 



