On the Work that may be gained during the Mixing of Gases. 311 



1,2,3... without omissions; (iv) partitions into the elements 

 1, 2, 3 . . . in which each must appear twice at least, except the 

 largest part, which need only appear once, but may appear any 

 number of times ; (v) the partitions described in the last para- 

 graph. 



Taking Euler's example of #=10, omitting for brevity the 

 signs connecting the parts, and writing, ex. gr., 1 + 1+2 + 2 + 4 

 as 1 2 2 2 4, the partitions of the different kinds are for (i) l 10 , 1 7 3, 

 1 5 5, 1 3 7, 19, 13 3 , 1 4 3 2 , 1 2 35, 37, 5 2 ; for (ii) 10, 19, 28, 37, 46, 

 127, 136, 145, 235, 1234; for (iii) I 10 , 1 8 2, 1 6 2 9 , 1 4 2 3 , 1 2 2 4 , 

 1 5 23, 1 3 2 2 3, 12 3 3, 1234; for (iv) 10, 1 2 8, 1 4 6, 1 6 4, 1 8 2, 3 2 4, 

 2 2 6, 1 2 2 2 4, 1 4 2 3 , 2 5 ; for (v) 10, 19, 127, 12 2 5, 12 3 3, 37, 28, 

 136, 1234, 24 2 . 



Putting # = 10 in (3), we have 

 1+P(1,2)4 + P(1,...4)3 + P(1,... 6)2+1 = 1 + 3 + 3 + 2+1; 

 while (5) gives P(l, 2)9 + P(l, 2,3,4)4 = 5 + 5. There is a for- 

 mula implicitly given by Euler( Opera minora collecta,vo\. i. p. 93) 

 which may be noticed here for the sake of completeness, viz. 



P(l, 3, 5 . . .)# = P(1, 2, 3 . . .)#- P(l, 2, 3 . . .){x-2) 



-P(l, 2, 3 . . .)(*— 4) +P(1, 2, 3 . . .)(«?- 10) + &c, 

 the terms being after the first alternately negative and positive 

 in pairs, and the general term being P(l, 2, 3 . . .) (x— 3n 2 ±Sn). 

 This formula would, of course, be quite inappropriate for the 

 calculation of P( I, 3, 5 ... x); for # = 10 it gives 

 42-22-11 + 1 = 10. 

 Cambridge, March 10, 1875. 



XXXIV. On the Work that mag be gained during the Mixing of 

 Gases. By Lord Rayleigh, M.A., F.R.S.* 



THE well-known fact that hydrogen tends to escape through 

 fine apertures more rapidly than air enters to supply its place, 

 even although the advantage of the greater pressure may be on 

 the side of the air, proves that the operation of mixing the 

 two gases has a certain mechanical value. In a common form 

 of the experiment a tube containing hydrogen and closed at the 

 upper end with a porous plug of plaster of Paris stands over 

 water. In a short time the escape of hydrogen creates a partial 

 vacuum in the tube, and the water rises accordingly. Whenever 

 then two gases are allowed to mix without the performance of 

 work, there is dissipation of energy, and an opportunity of doing 

 work at the expense of low temperature heat has been for ever 

 lost. The present paper is an attempt to calculate this amount 

 of work. 



* Communicated bv the Author. 



