244 



NATURE 



\yuly 26, 1877 



%=.T - M [Uu + Vv+ Ww +p (wy - vs) +q {uz - 



■wx) + r (vx - uy) + C M, 

 the distribution of velocities will still be a permanent one. 



In this expression the quantities U, V, IV, p, q, r, may 

 have any values provided they arc the same for the 

 whole system of bodies, but C may be different for 

 different bodies, because it is multiplied by the mass 

 of the body, which is invariable. But we arrive at the 

 same expression by substituting in T for ti, v, and "W the 

 quantities — • 



« - U -\- qx - ry 

 V — V -\- rx - pz 

 'M - W +py - qx. 

 or, in other words, by substituting for the absolute velo- 

 cities of the bodies their velocities relative to a system of 

 axes moving in the most general manner possible ; that 

 is to say, the components of velocity of the origin being 

 U, V, IV, and the components of the velocity of rotation 

 being/, q, r, and at the same adding to Tthe quantity 



i M { {qz - vyf + {rx - Z^)' + (/j - qxf } - C , 

 which depends on the co-ordinates only and not on the 

 velocity of the body. 



We now see that the most general case of permanent 

 distribution is when the system of bodies is contained in 

 a vessel of invariable form which moves with constant 

 velocity along a screw, that is to say, in which one point 

 is moving along a straight line with constant velocity, 

 while the vessel rotates about an axis passing through 

 this point with constant angular velocity. 



When there is rotation, we must subtract from the 

 potential energy a term depending on the co-ordinates, 

 which shows that the rotation produces an effect similar 

 to that of a centrifugal force at right angles to the axis 

 of rotation. 



Returning to the general expression for the number of 

 molecules in a group, we may make it yield us informa- 

 tion of other kinds. Thus, if we wish to know the density 

 of a particular gas at any given point in the mixture, we 

 have only to make the limits of the group those of an 

 element of volume, and we find the density proportional to 



i?" '", where X is that part of the energy of a single 

 molecule which is due to external forces, such as gravity. 

 In the case of gravity, x is equal to m s; z, where tn is the 

 mass of a molecule, g the intensity of gravity, and z the 

 height. This leads to the ordinary expression for the 

 density of a gas of uniform- temperature in a vertical 

 column, and it shows that in the ultimate distribution of 

 a mixture of gases the density of each gas diminishes 

 with the height according to its own law, that is to say that 

 of the heaviest gases diminishes most rapidly, so that the 

 proportion of the heavier gases diminishes with the height. 



This law of the distribution of gases was asserted by 

 Dalton as a consequence of his theory of gases, and 

 numerous experiments have been made on air collected 

 at different heights in the atmosphere in order to 

 detect a difference in their composition, but we cannot 

 say that such a difference has as yet been satisfactorily 

 established. 



The atmosphere, in fact, is eminently unfitted for 

 testing the theory of the ultimate state of a mixture in 

 equilibrium, for the inequalities of temperature in so large 

 a body of gas produce powerful currents which continu- 

 ally carry masses of the mixture from one stratum into 



another. This tends to produce a uniformity of com- 

 position and a variationof temperature which are both of 

 them contrary to our theory of the condition of equilibrium, 

 and which seem to favour certain' other theories. 



Nor is the case much improved if, instead of the open 

 atmosphere, we substitute a mixture of gases contained in 

 a vertical tube. For in order to obtain a difference of 

 composition at the top and bottom of the tube large 

 enough for experimental verification, the tube must be 

 at least 100 metres high, and it would take more than a 

 year for the contents of such a tube to approximate by 

 one half to their final distribution. In the'mean time the 

 slightest difference of temperature in the sides of the tube 

 would produce currents which would tend to equalise the 

 composition of the mixture. To verify the other result of 

 our theory — the uniformity of temperature in the ultimate 

 state of a vertical column — would be attended with still 

 greater difficulties. 



But it would be quite within the powers of experimental 

 methods to verify the law of distribution of a mixture of 

 gases in a rotating vessel. Let two bulbs be connected 

 by a wide tube, say 10 cm. long, and let them be filled 

 with equal volumes of hydrogen and carbonic acid, well 

 mixed together. Let this apparatus be placed on a 

 whirling machine, so that one bulb shall be close to the 

 axis, while the other is moving at the rate of fifty metres 

 per second. The same degree of approximation to the 

 final state, which would take years irv, the long tube, will 

 be effected in minutes in this small apparatus, and the 

 proportion of carbonic acid to hydrogen will be about ,J;j 

 greater in the bulb furthest from the axis. 



The clear demonstration of this proposition given by 

 Mr. Watson is of great scientific value, for almost every 

 one of those who have attacked the question with 

 insufficient methods of investigation have come to the 

 conclusion that the temperature would diminish in a 

 vertical column as the height increases ; and those who 

 regard gaseous diffusion from a chemical rather than 

 a dynamical point of view would probably expect the 

 composition to be uniform at all heights. 



But the profound scientific value of this proposition 

 becomes more manifest when we make use of it in 

 establishing the definition of temperature and the law of 

 volumes of gases. 



In Prop. II. of this book, which corresponds to the 

 original form of the theorem, as I gave it in the 

 Philosophical Magazine, January, 1 860, two sets of 

 spheres are completely mixed up together in the same 

 vessel, and it is proved that the average kinetic energy of 

 a sphere is the same for either set. We may then assert, 

 as I did, that the two gases are at the same temperature 

 because they are thoroughly mixed together. But this 

 assertion has no scientific meaning, because we cannot 

 test its truth by putting a thermometer first into the one 

 gas and then into the other. 



But if we now call to our aid a system of forces acting 

 on the molecules and tending to fixed centres, we may 

 obtain a result capable of experimental verification ; for 

 though we are not acquainted with natural forces acting 

 exclusively on one kind of gas, we can calculate the 

 effects of such forces. 



Let us assume, then, that the forces are such that the 

 potential energy of a sphere of the set A' is much greater 



