382 Mr. W. Sutherland on Thermal 



by our last expression be constant for a given gas and all 

 values of jt? 3 , which was the experimental result obtained by 

 Reynolds ; moreover if we wish to compare different gases, 

 as at a given temperature v is proportional to J//n y we see 

 that the constant time of transpiration for each gas ought to 

 to be as the square root of its molecular mass, which is Graham's 

 well-known experimental discovery verified by Reynolds. 



This digression into the properties of a gas in spaces where 

 the linear dimensions are small compared to the free path has 

 been made as an appendix to our consideration of slipping in 

 order to clear up the limiting conditions towards which we 

 tend in treating of high vacua. We can now return to 

 thermal transpiration as we left it at (4). To secure no total 

 flow on account of u along a tube of radius R we are to have 



B = 7r 1 y\V(l + mR)/Sv = '7r'Rhi= -~irB?v\(n'/n + v'/v)/6; (12) 



but p — nmv 2 !?)^ so that p , /p = n / /n + 2v , /v, and then 



/R 2 (l -r-4f/R)/877= -v\(p'/p-v'/v)/6. . (13) 



Now with the methods of approximation here employed 

 7] = nmv\/4: and p = nmv 2 /3, so that r) = 3\p/4;V, and then 



£{§(l + 4 ? /R) + l}=4 . . . (14) 



As £=a\ the coefficient of p'/p is a function of only R/X, and 

 therefore the controlling influence of the whole phenomenon 

 of thermal transpiration is this ratio of R to X. 



If the molecules are smooth., perfectly restitutional forceless 

 spheres r]=r J0 v/v , where rj and v are the values of rj and v 

 at 0°C. ; but with the molecules of the natural gases, on account 

 of molecular force, the function which expresses ?; in terms of 

 v is more complicated (see " Viscosity of Gases and Molecular 

 Force," Phil. Mag. [5] xxxvi.). But for present purposes it 

 will suffice to use the simple relation just given by which we 

 can express the last differential equation in terms of p and v 

 as the only variables thus 



dp/ dUWp . 3ARk \ Id _i_ n n _ . 



M16170V vov* J pdv ' ' { Dj 



itten as 



^(% + »W| log i> = o. 



dv \ v 4 v 2 J dv & v 



which can be written as 



Let « and £ stand for - ^ + -^ (D 2 -2C)*; 



