376 Mr. W. Sutherland on Thermal 



number of molecules colliding in a second in dB is nvdB. It 

 is not worth while to take account of Maxwell's law of velo- 

 cities ; and all molecules will be supposed to have the average 

 velocity v and travel the mean free path X between two 

 encounters, so that v=v/X. But we must take account of the 

 variation of X with direction ; for a molecule travelling from 

 a particular point has a longer path in the direction of dimi- 

 nishing density and a shorter path in that of increasing 

 density, with a maximum parallel to the axis in one direction 

 and a minimum in the other ; while the path at right angles 

 to the axis is the mean of the maximum and minimum, and 

 is indeed the mean path X of all molecules leaving that point. 

 Let X m be the maximum value there ; then it is equal to the 

 minimum at distance X m along the axis of the tube, and must 

 therefore be equal to the mean value at distance X^/2 ; thus, 

 then, X m = X -f X m dX/dx 2, or X m = X + X dX/dx 2 = X ( 1 + \'/2) . 

 On the same principle, the free path of a molecule that leaves 

 the point in any direction so that the projection of its path on 

 the axis of the tube is x, has a value \+ t v\'/2. Of the num- 

 ber nvdB /X of molecules that in unit time have a collision in 

 dB, the fraction that cross a plane at any distance is found 

 by drawing from the centre of dB as origin the surface whose 

 polar equation is p = X + xX f j2, and estimating the solid angle 

 subtended at the origin by the segment of this surface cut off' 

 by the plane, supposed to be at distance a?, as a fraction of 47r. 

 This is the required fraction, namely {l — x/(X-\-xX / /2)}/2, or 

 (l-^/X + ^ 2 X72X 2 )/2 nearly. 



Thus the number of molecules colliding in dB and crossing 

 the plane before colliding again is in unit time 



nv dB(l- x/X - x 2 X'/2X 2 ) /2X, 



in which we have changed the sign of X' so as to transfer the 

 origin from dB to the plane. Now dB may be taken as Adx 

 where A is the area of section of the tube ; the total number 

 crossing the plane from the tube on one side of it in unit time 

 is the integral from to X 1 of Anv(l-xlX-x*X f /2X*)dx/2X, 

 where X : is the maximum free path at such a distance from 

 the plane that a molecule after colliding there and travelling 

 perpendicular to the plane collides again just at the plane. 

 Now X=c/n, where c is a parameter depending only on the 

 size of the molecules ; thus the number is 



( An 2 v(l-^/c~-^ 2 .i' 2 X72c 2 )^/2c . . . (1) 



Now if r? and v are the values of n and v at the plane 



