498 On the Separation of Gases by Diffusion. 



u induction ") of n sets of operations by 



(16) 



/r + s\ n 



When we take account of the reciprocal character of r and s, 

 this may he written 



the number of parts into which the original quantity of gas is 

 divided being n + 1. If n is even, the largest part, corre- 

 sponding to the middle term, has the original composition*. 



It is to be observed, however, that so far as the extreme 

 concentration of the less diffusive constituent is concerned 

 these complex operations are entirely unnecessary. The same 

 result, represented by {\) n [ r n will be reached at a single 

 operation by continuing the diffusion until the residue is 

 reduced to {^) n of the original quantity, when its composition 

 will be that denoted by r n . And even as regards the extreme 

 member at the other end in which the more diffusive con- 

 stituent preponderates, it will be evident that the operations 

 really required are comparatively simple, the extreme member 

 in each row being derived solely from the extreme member of 

 the row preceding f. 



If we abandon the supposition, adopted for simplicity, that 

 the gas is divided into equal parts at each operation, we may 

 still express the results in a similar manner. If p, a be the 

 fractions retained and transmitted, then p + a = l, and in place 

 of (15) we get r=p k ^ 



The relation between r and s is 



pr + <rs=l ; (19) 



and the various portions into which the gas is divided after n 

 sets of operations are represented by the various terms of the 



expansion of (pr + <")", (20) 



the Greek letters and the numerical coefficients giving the 

 quantity of each portion, and the Roman letters giving the 

 quality. But it must not be forgotten that this theory all along 

 supposes the difference of diffusivities to be relatively small. 



* There is here a formal analogy with the problem of determining the 

 probability of a given combination of heads and tails in a set of n tosses 

 of a coin ; and the result of supposing n infinite may be traced, as in the 

 theory of errors. 



t Possibly a better plan for the concentration of the lighter constituent 

 would be diffusion along a column of easily absorbable gas, e. g. C0 2 . 

 The gas which arrives first at the remote end is infinitely rich in this 

 constituent. 



