Transpiration and Radiometer Motion. 383 



then the integral of this is 



-logip/v*-*) - — \ O g( p / v *-0) + °^,\ogp/v= constant, 



or with suffixes 1 and 2 to indicate the ends of the tube, 



2i li, g^^-^£k^8 + ^r ,og ^r =0 - (16) 



But this is a very awkward form of result for comparison 

 with the experimental data, and we shall be better served if 

 content with an approximate solution of the differential equa- 

 tion obtained by putting p' = dp/dx = (p 2 —pi)/l and v = 

 (^2—^i)/h '2p=p2±Pi, 2t7=tf s + v 1 ,.tlms 



P2-P1 f 9RV(p 2 4-p!) 2 6aB J v (p 2 +p } ) +1 "\ _ y 2 -i\ ,j~ 



This solution brings out at once the important point that with 

 v 2 and v x fixed, that is to say, the temperatures of the two 

 ends constant, there is a certain mean pressure (p 2 + pi)/2 for 

 which p 2 ~pi has a maximum value; applying the usual con- 

 dition for a maximum, we find that p 2 —p\ is a maximum 

 when p 2 +/>i = 2t; (?: 2 + r 1 ) 2 /3Ri' . Before proceeding to test 

 (17) by Reynolds's experiments, we may remark that if the 

 mean pressure (p 2 +pi)/2 is made so small that R/A is negligible 

 in comparison with unity, then in (14) p f /p = v'/v, that is 

 Ps/pi ==v 2/ v u a resn lt in accordance with the following common- 

 sense argument that when the mean path of a molecule is a 

 large multiple of the radius of the tube, the molecules of the 

 tube have practically no influence on one another ; and the 

 number that wander in at one end during unit time being 

 n^j/4 and at the other n 2 v 2 /i, thermal transpiration will con- 

 tinue till these are equal, that is till n 1 v 1 = n 2 v 2 or p- i /v- [ =p 2 /v 2 . 

 So far, our theoretical treatment has related to cylindrical 

 tubes, while in Reynolds's experiment the passages through 

 which the gases transpire are the irregular chains of cavities 

 in a porous plate ; now to a first approximation these irregular 

 cavities may be replaced by uniform tubes whose sectional 

 area is equal to the average section of the cavities, but it is 

 obvious that a better approximation to the natural cavities 

 would be a succession of frustra of cones of length L and 

 radii R x and R 2 at the end sections. The thermal transpiration 

 through such a frustrumcan be readily established from (15), 

 for taking the origin of coordinates in one end and in the 

 axis, then the radius at distance x along the frustrum is 

 R = R x + c#, where c is a constant : thus for the fall of pressure 



