Following are a few of the results obtained by the mathematical study of such molecules as are defined above. It is evident, in the first place, that the velocities of the various mole cules will not all be equal; for even if such equality existed at any given instant, it would be quickly destroyed by the inter-molecular col lisions. Maxwell investigated the distribution of velocities that must subsist in a gas com posed of such molecules and gave a formula by which it is possible to calculate, at any given instant, the number of molecules that have velocities greater than, or less than. any assigned velocity. Thus if the total number of molecules present be taken as unity, the number having a velocity less than the average velocity is 0.533; the number having a velocity less than one-half the average velocity is 0.112; the num ber having a velocity less than twice the average velocity is 0.9829; and the number having a velocity greater than four times the average velocity is 0.0000000074. It appears, therefore, that although any velocity whatever is theo retically possible (so far as Maxwell's formula is concerned), the incessant collisions bring about a sort of averaging which is effective enough to ensure that an almost vanishingly small proportion of the whole number will be actually moving with a speed as great as four times the average. The number having higher velocities falls off with still more remarkable rapidity; for example, the formula shows that less than one molecule in 10" will be moving with a speed as great as 10 times the average. When two or more different kinds of molecules are simultaneously present, all the molecules in any one set being exactly alike and very numerous, and every molecule being hard smooth, small, spherical and perfectly elastic, Maxwell found that the different sets will mix with one another uniformly, and that the veloci ties in each set will be distributed precisely as though the other sets were not present. The average velocity in each set will be different, however, from the average velocity in every other set, the set in which the molecules are heaviest having the smallest average velocity. In fact, the velocities, in such a case, will be such that the average kinetic energy of a molecule of one set will be precisely equal to the average kinetic energy of a molecule of any other set.
Some of the mathematical difficulties that appeared almost insuperable to Maxwell have been partially overcome • by other mathemati cians, and, largely owing to the labors of Boltz mann, we now have a far more general form of the kinetic theory of gases. Before stating the nature of the generalizations that Boltz mann effected, it is necessary to offer a short explanation of the expression "degrees of free dom.° A mathematical point is completely de fined when its three co-ordinates are given; it can move by the variation of any one of these three co-ordinates, while the other two remain constant. Such a point is therefore said to pos sess three "degrees of freedom.° A rigid body in space similarly has six degrees of freedom. Three co-ordinates must be given in order to fix the position of some one of its points — say its centre of gravity i and it may also have three independent rotations about three independent axes passing through the point so fixed. If the rigid body is not free to rotate, or if (as in the case of the smooth spherical molecules imag ined by Maxwell) there is no force acting which tends to produce rotation, the number of de grees of freedom may be considered as reduced to three, the three co-ordinates of the centre of gravity being then sufficient to define the state of the body completely. In particular, a mole
cule shaped like a dumb-bell may be considered to have but five degrees of freedom, if it is so smooth that collisions cannot set it in rotation about its axis of symmetry. The number of degrees of freedom of a rigid body is six, in the most general case; but if two or more rigid bodies be joined together by hinges, or by any other analogous mode of connection that will allow of relative motion between the compo nents, the number of degrees of freedom of the system so formed becomes greater than six. Thus a system composed of N straight rods, connected together by flexible joints at their ends, has (2N + 3) degrees of freedom.
Boltzmann's form of the kinetic theory may now be stated as follows: Let there be a gas composed of any number of sets of molecules, such that the molecules belonging to each set are exactly like one another, though a molecule belonging to one set may be totally unlike a molecule belonging to another set. Let these molecules have any number of degrees of free dom (which number of degrees may be dif ferent in the different sets), and let them be acted upon by parallel forces (such as gravity), or by forces tending toward fixed centres, or by internal forces (that is, forces acting within the individual molecules, between their parts). Let all the molecules be very small in com parison with the total space they occupy, so that the chance of their colliding three or more at a time is practically nothing. Moreover, let them be very numerous and let them be per fectly elastic, and let them be smooth, so that when they collide the only force tending to make them rotate is that due to normal impact. Let them be set in motion among one another with any distribution of velocities; and let them be hard, but not infinitely so — the force called into play collision being very great, but not infinite (as it would be if the hardness were infinite) ; and let the duration of a collision be exceedingly short, yet not neces sarily zero. Then Boltzmann reaches the fol lowing conclusions: (1) After a short time, the law of distribution of positions and velocities in each set of the molecules will be precisely the same as it would be if all the other sets were absent; so that each set behaves as a vacuum to all the rest, so far as the distribution of ve locities and the density of aggregation of the molecules in any given region are concerned. (2) The law of distribution of the velocities of translation in each set is the same as that de duced by Maxwell for spherical molecules. (3) The average kinetic energy of translation of the molecules of any one set is equal to the average kinetic energy of translation of any other set.. (4) The total kinetic energy of each set of molecules (including that due to translation, rotation, etc.) is divided up equally among the different de grees of freedom of that set. This last prop osition is undoubtedly one of the most remark able ever enunciated with regard to molecules and it appears not to have met with unqualified acceptance among mathematicians, though there are many experimental facts. which tend to show that it is at all events a good approxima tion to the truth.