"336 Combinatorial Analysis. — The Foundations of a Nevj Theory. 



rate at whioh the planet would lose its atmosphere, since it takes 

 no account of molecules which describe free paths beyond the limit 

 ■and fall back again. To further exhibit the results in a tangible form, 

 the rate of flow is estimated by the number of years in which the 

 total amount of gas escaping across the critical surface would be 

 equal to the amount of the gas in a layer covering the surface of the 

 planet to the depth of 1 cm. This measure is independent of the 

 -actual quantity of the gas under consideration existing in the atmo- 

 sphere, since, if this quantity be increased, the rate of flow across the 

 critical surface and the amount of gas present in the surface layer 

 1 cm. thick will be increased in the same proportion. 



If a gas of molecular weight 2, such as helium, be supposed to exist 

 in the earth's atmosphere, the loss in question would occupy 3-5 x 10^^ 

 years at - 73° C, 3 x IQi^ years at 27°, 8*4 x lO^o years at 127° C, 

 6 X 10^ years at 227° C, and 222 years at 327° C. 



If we halve the absolute temperatures we have the conditions 

 applicable to hydrogen, the losses in question therefore taking place 

 in 8-4 X IQio years at -73° C, 6 x lO^ years at - 23° C, and 222 

 years at 27° C. 



For water vapour on Mars, the corresponding results are 1-2 x 10^^ 

 years at - 73°, 1-9 x IQi^ years at 27°, 2-4 x 10^ years at 127°, 

 4'3 X 10^ years at 227° and 106 years at 327°. 



These figures indicate that helium cannot practically escape from 

 our atmosphere at existing temperatures, nor can vapour of water 

 •escape from the atmosphere of Mars. A leakage may and undoubtedly 

 does take place which may appear considerable when estimated by the 

 number of actual molecules escaping, but it is wholly inappreciable 

 relative to the mass of gas left behind. 



At a future time I propose to examine the corresponding results, 

 based on the hypothesis that the atmosphere of a planet is distributed 

 according to the adiabatic instead of the isothermal law. 



^'Combinatorial Analysis. — The Foundations of a "NTew Theory." 

 By Major P. A. MacMahon, E.A., D.Sc, F.E.S. Received 

 March 19,— Bead April 5, 1900. 



(Abstract.) 



The object of the paper is to exhibit the processes of the infinitesimal 

 •calculus and of the calculus of finite difl'erences as combinatorial 

 processes. A large class of problems can be dealt with by designing 

 on the one hand a function, and on the other hand an operation, in 

 such wise that when the operation is performed upon the function a 



