460 MATHEMATICAL APPENDICES 54* 



would hold good, if the composition of the mixture determined by 

 NI and N 2 depends only on a single coordinate x in addition to 

 the time t. In that case N } and N, 2 may be replaced by the 

 partial pressures p l and p 2 , which each gas would exert were it 

 alone. 



But if the coefficient of diffusion D varies with NI and N z , and 

 consequently with x, the equations become 



d dNi\ &Nt dD dN, 



= d ( D dNi\ = 

 dx\ dx J 



dt dx\ dx dx 2 dx dx 



dN 2 ^ d ( D dN*\ __ j) , 



dt dx\ dx J dx z dx dx ' 



But now 



dD _ dD_dNi , dD dN 2 

 dx dN, dx * dN 2 dx' 



or, since the numbers NI and N 2 are connected by the equation 



N,+N 2 = N, 



in which N denotes the whole number, independent of x and t, of 



the particles in unit volume, 



AD = (dD _dD\dN ( dD dD\dN 2 

 dx " \dN l dNj dx " \dN 2 dNJ dx' 



The differential equations therefore take the form 



D d^N 1 fdD _ dD 



dt dx 2 \dN l dNj \ dx 



d-N 2 . J-LI^ _i 



dt dx* ' 



in which the square of the first differential coefficient with respect 

 to the coordinate is involved as well as the second. 



If the diffusion tube is long enough for the rate of change of 

 the partial pressure of a gas to be very small, then (dN^dx) 2 and 

 (dN 2 /dxY are small magnitudes which may be neglected. The 

 equations then retake the simpler form 



AN* 



dt dx 2 ' dt dx 2 ' 



but they contain a factor D which depends on N l and N 2 , and 

 thus varies with x and t. If we are to assume solutions of these 

 equations of the usual form, we shall have so to arrange our 

 experiments that NI and N 2 , and therefore also p l and p 2 , may 

 alter with time and place in only a very slight degree. 



