with Remarks upon the Mechanical Theory of Heat. 87 

 sponding magnitude for a layer of the thickness 8, we have only 

 to multiply the previous fraction by -, that is, 



and if this magnitude be subtracted from 1, the difference 

 represents the free portion of the plane as a fraction of the whole 

 plane. 



Hence the probability that the point will pass through our 

 plane, or, which comes to the same thing, through a layer of 

 thickness 8, without obstruction, is determined by the equation 



W. = l-^'8; (3) 



and on comparing this expression for W^ with that given in 

 equation (2), we find that 



_ ""P 



a:= 



(4) 



\3' 



and hence the general equation (1) is transformed into 



'^p' - 

 W = e-r»" (5) 



(6.) By means of this equation we can now determine the 

 mean value of the path which the point has to traverse before 

 it meets with a sphere of action. 



Let us suppose that a great number (N) of points are thrown 

 through space in one direction, and let us suppose the space to 

 be divided into very thin layers perpendicular to the direction 

 of motion ; then a small number of the points would be detained 

 in the first layer by the spheres of action, another lot in the 

 second, another in the third, and so on. If, now, each of these 

 small numbers be multiplied by the length of path, the products 

 added, and the sum obtained divided by the whole number N, 

 the quotient will be the mean length of the path which we 

 seek. 



According to equation (5), the number of points which either 

 reach or pass the distance x from the commencement of the 

 motion is represented by 



and accordingly the number which reach or pass the distance 

 x^-dx'\^ expressed by 



