the beginning of the first of these parts the time zero; the beginning of the second t1, the beginning of the third t2, &c. ; the end of the last of the n parts t. After the whole time has elapsed, let another series of times of length begin. Denote the end of the first part after t, by t,+1; the end of the next following part t+ &c. Assume for a moment that we have n separate vessels, all exactly similar to the one containing the gas; that each of these n vessels contains the same gas, and that the motion of the gas is the same in each. The beginning, however, is different. For example, let the gas in the second vessel at time zero be in the same condition in which the gas of the first vessel is at the time t,; in the third vessel let the gas at the time zero be in exactly the same condition as it is in the first vessel at the time të, and so on. We have now in the different vessels all the different states of the gas existing simultaneously which in the first vessel exist successively during the whole time interval ✪. The probability dw that the co-ordinates and momenta of a molecule may lie between the limits a and a+da, b and b+db... p and p+dp, q and q+dq.... (1) can be defined in two ways. If we consider a single vessel containing gas, we must observe it for a long time; if be the fraction of the time during which the co-ordinates and momenta of a molecule lie between the limits (1)—which we shall call the condition (1)—then / is the probability required. The limits (1) differ only infinitesimally from one another. No two molecules of the same gas can be in the condition (1) at the same time. On the other hand, if we consider the above series of n vessels at any single instant of time, we can define the probability dw to be dz/n, where dz is the number of vessels in which a molecule is in the condition (1). Evidently dw will have different values for different values of the co-ordinates and momenta. It will also be proportional to the differentials da, db... We may therefore put To find the condition for a stationary state we may consider one gas at successive instants, or the series of vessels at one instant. In the first case the values of dw for the stationary state will be the same, whether we consider the gas from time 0 to time t, or from time t, to time +1, or in general from t to t+ Evidently the converse is true; that is, if du has the same values for all these cases, the state is stationary. By the second method we must remember that at the time O we have in our n vessels all the states which appear in the first case from time 0 to time t,; at the time t, we have in these vessels all the states which appear in the first case from t1 to t+.... The above statement, T that has the same values in all cases, whether we consider the time from zero to t,, or from t, to t+1 or from t to t+, becomes in this second case identical with the statement that dz/n has the same value, whether we consider the n vessels at time zero, or time t1, or time t2, &c. That is, since the difference between t1 and zero, t, and t,... can be made infinitely small, the above statement amounts to saying that for the stationary state dz/n has the same values at all times. We shall next prove that d/n has this property under the following conditions: We define a free molecule to be one which is not acted upon by any other molecule. For each free molecule let the values of ƒ (a, b . . . p, q . . .) He- at time 0, where g is the total energy of this molecule; H and are constants; therefore at time 0 the number of vessels in which a free molecule is in the state (1) is Similarly the number of vessels in which a free molecule appears with co-ordinates and momenta between a' and a' + da', b' and b'+db' . . . p' and p' +dp', q' and q' +dq' . (condition 4) is ndW'=nHe-ho' da'db'... dp'dq' ... ... (4) where g' is the total energy of this second free molecule. Finally, the number of vessels in which one free molecule is in condition (1) and a second one in condition (4) is n dW dW'=nH2e-h(g+g') da db Let a", b', . . . a'", b'"', be values of the co-ordinates such that a molecule with the former co-ordinates acts on or encounters a molecule with the latter co-ordinates. And let us assume that at the time 0 the number of vessels which contain a pair of molecules whose co-ordinates and momenta respectively lie between a" and a" + da", b" and b" + db". . . . · p" and p" + dp", q" and q" + dq" a""' and a"" + da""', b'"' and b'"'+db"", """" and ""'+dp"", "" and (6) is ... da"" db"" ... dp" dq""... nH2e-h da''db' . . . dp"dq" (7) where ƒ is the total energy of the two molecules. We proceed to prove that a stationary state is defined by these formulæ. Consider a duration of time t long enough to permit of encounters between a finite number of molecules, but not so long as to permit of many molecules colliding more than once. We must demonstrate that after this time t, the number of vessels in which the state of a molecule lies between certain limits is exactly the same as before this time. We distinguish between four kinds of molecules :: (i) Molecules which are free at the beginning and at the end, and during the whole time t. For any of these molecules let the co-ordinates and momenta lie at the time 0 between A and A+dA, B and B + dB,... P and P+ dP, Q and Q+dQ... (8) and at the time t between a and a + da, b and b + db, . . p and p+dp, q and q + dq... (9) Let G and g represent the energy of such a molecule at the times 0 and t respectively; G, g being equal. According to equation (3) the number of vessels in which at the time 0 the co-ordinates and momenta of a molecule lie between the limits (8) is nHe-GdA dB,... dP dQ. . . . But, by hypothesis, the co-ordinates and momenta of these same molecules lie between the limits (9) at the time t; hence the above expression gives also the number of vessels in which, at the time t, co-ordinates and momenta of a molecule lie between the limits (9). But we have G=g, and by a well-known theorem (cf. Watson, Kinetic Theory of Gases,' 2nd edition, p. 22) da db... dp dq .= dA dB . . . dP dQ Therefore the number is equal to nHe-1g da db... dp dq . . . But this last expression gives at the time 0 the number of vessels for which the co-ordinates and momenta of a molecule lie between the limits (9) (according to formula 3). We see that this number remains constant during the time t; and since the same is true for all values of a, b,. P, q, the theorem holds good for all molecules of the first kind. (ii) We call all those molecules 'molecules of the second kind' which are free at the time 0, but which are in process of encounter at the time t. For a pair of such molecules let the co-ordinates and momenta lie at the time 0 between the limits and ... ... ... A, and A, +dA,, B, and B, + dB, P, and P, + dP,, Q, and Q, + dQ1 respectively, and at the time t between the limits } (10) a, and a,+ da, b, and b, + db, . . . and a, and a + du, b2 and b2 + db2. P2 and P2+ dp, 12 and 12+ dq2 respectively. Because these molecules were free at the time 0, the number of vessels in which at time 0 a pair of molecules fulfils the condition (10) is, according to formula (5), nH2e-(G,+G) dAdB1 dPdQ,... dAdB2 . . . dPdQ2 ・ ・ ・ G, and G, are the energies of the molecules at the time 0. But the above-mentioned vessels are identical with the vessels for which at the time t a pair of molecules fulfil the conditions (11). The number of the last kind of vessels is therefore also given by the above expression. It is easily seen that this expression is equal to nH2e-da,db1...dp.dq da,db... dp2dq . . . ... where f is the whole energy of the two molecules at time t. Comparison with formula (7) shows that the last formula gives also the number of vessels in which at time 0 a pair of molecules fulfilled the condition (11). Therefore the theorem also holds good for the molecules of the second kind. (iii) Molecules which are in process of encounter at time 0, but are free at time t; (iv) Molecules which are free at times 0 and t, but which have been encountered by another molecule between these two instants of time. It is easily seen that our theorem can be proved in the same way as before for every pair of molecules of the third or fourth kind. To calculate the mean vis viva T of a molecule we put x1 =k11p+k122 + ..., x2=k21p+k229. . ., &c. The coefficients k may be chosen to be functions of the co-ordinates such that T acquires the form (m,x,2+m, x2 + ... mx), where f is the number of degrees of freedom of a molecule. The probability that for a molecule the co-ordinates and the values of x may lie between a and a+da, b and b+db... x, and x1+dx1, x2 and x2+dx2... is, according to formula (3), where He-h(+mx2) Dda db dx dx2... ... It is evident that each momentum, and therefore, also, each of the variables x, can assume all values from -∞ to +∞. We easily obtain the value 1/2h for the average value of each term of the form mx2. Therefore the average vis viva of a molecule is ƒ/2h, the average vis viva of the centre of mass of a molecule is 3/2h, and the ratio of these two quantities is f: 3. The Best Methods of Recording the Direct Intensity of Solar Radiation.Tenth Report of the Committee, consisting of Sir G. G. STOKES (Chairman), Professor A. SCHUSTER, Mr. G. JOHNSTONE STONEY, Sir H. E. ROSCOE, Captain W. DE W. ABNEY, Mr. C. CHREE, Mr. G. J. SYMONS, Mr. W. E. WILSON, and Professor H. MCLEOD. (Drawn up by Professor MCLEOD.) VERY little has been done with Balfour Stewart's actinometer during the past year. It will be remembered that in the last Report it was stated that an attempt had been made to replace the thermometer by a thermocouple of copper and iron. From the preliminary experiments it appears that this arrangement is extremely sensitive, and using the instrument as a dynamical actinometer, in which the rate of change of temperature is recorded, a complete observation may be made in from two to three minutes. It was mentioned last year that a D'Arsonval galvanometer had been tried; the Committee have now purchased an Ayrton-Mather galvanometer specially wound for thermo-electric currents: this instrument has been examined by Professor Ayrton, and to him and his pupil, Mr. Arnold Philip, the Committee are indebted for much useful information. The instrument is not yet in a very satisfactory condition, for, in order to make it sufficiently sensitive, the suspending wire has to be unusually fine, and it takes a permanent set, which causes an alteration of zero. Endeavours are being made to overcome this inconvenience. The thermocouple of copper and iron does not give currents quite proportional to the difference of temperature, and it might be preferable to replace the iron by some other metal or alloy. Copper is, of course, one of the essential metals, and it appears difficult to find any other material to replace the iron which will give proportional currents of sufficient strength to be useful. The Committee ask for reappointment and for the unexpended portion of the grant. Underground Temperature.-Twentieth Report of the Committee, consisting of Professor J. D. EVERETT, Professor Lord KELVIN, Mr. G. J. SYMONS, Sir A. GEIKIE, Mr. J. GLAISHER, Professor EDWARD HULL, Professor J. PRESTWICH, Dr. C. LE NEVE FOSTER, Professor A. S. HERSCHEL, Professor G. A. LEBOUR, Mr. A. B. WYNNE, Mr. W. GALLOWAY, Mr. JOSEPH DICKINSON, Mr. G. F. DEACON, Mr. E. WETHERED, Mr. A. STRAHAN, and Professor MICHIE SMITH. (Drawn up by Professor EVERETT, Secretary.) THE Committee were appointed for the purpose of investigating the rate of increase of underground temperature downwards in various localities of dry land and under water. The nineteenth Report contained the results of observations taken in 1891 by Mr. Hallock, of the Smithsonian Institution, at depths extending to 4,462 feet in a nearly dry well at Wheeling, Virginia. Mr. Hallock, who now dates from Columbia College, New York, has recently furnished the Secretary with printed copies of a paper, contributed by him to the American Association for the Advancement of Science last year, containing further observations in the well, made at the expense of the U.S. Geological Survey. When the observations of 1891 were finished, an oak plug was driven into the top of the casing to protect the hole. In July 1893 the plug was withdrawn, and the well, instead of being dry as before, was found to be full of fresh water to within 40 feet of the top. This water is believed to have leaked in at the lower end of the innermost casing-that is, at 1,570 feet below the surface. By means of inverted Negretti maximum thermometers, protected against pressure by sealing them in stout glass tubes, careful observations were taken at various depths from 1,586 feet to 3,196 feet, two thermometers being employed to check one another at each depth. The results were practically identical with those obtained two years previously, when the well was full of air, the greatest certain difference being only one-fifth of a degree. An obstruction at 3,200 feet prevented observation at greater depths; but this obstruction will probably be removed, the well pumped dry, and the drilling continued. In making the observations, four thermometers were lowered at a time, two of them being in an iron bucket 3 feet long and 3 inches in diameter at the end of the wire, and the other two in an open wire frame 260 feet from the end of the wire, the diameter of the bore being just under 5 inches. The temperatures at 103 feet, 206 feet, and 300 feet were also observed with suitable thermometers, the temperature at 103 feet being 52°-53, which is 1°.2 higher than the true temperature of the soil at that depth, as determined by other observations in the immediate neighbourhood. The smallness of the disturbance of temperature by convective circulation in this well, both when dry and when filled with water, is very remarkable, and renders the well specially suitable for determinations of the increase of temperature downwards. The Committee have to record with deep regret the loss of their valuable member, Mr. Pengelly. |