MAXWELL: THE DYNAMICAL THEORY OF GASES so that the probability is independent of that is, all directions of Let N be the whole number of particles. Let x, y, z be the com ponents of the velocity of each particle in three rectangular direc- Prop. III. Given the direction and magnitude of the velocities of tions, and let the number of particles for which x lies between x and wo spheres before impact, and the line of centres at impact; to find x+ dx, be N/(x)dx, where f(x)is a function of x to be determined the velocities after impact. The number of particles for which y will be Nf)dy tween y and y+ dy Let 04, OB represent the velocities before impact, so that if there e number for which z lies between z and had been no action between the bodies they would have been at A and B at the end of a second, Join AB and let g be their centre of t dz will be Nf(z)dz, where f always stands for the same function. Now the existence of the velocity x does not in any way affect that of the velocities y or z, since these are all at right angles to each other and independent, so that the number of particles whose velocity lies between x and x t dx, and also between y and y dy, and also between z and z+dz, is Nf(ruf(z)dx dydz If we suppose the N particles to start from the at the same instant, then this will be th (dx dy dz)after unit of time gravity,the position of which is not affected by their mutual action d to unit of volume will be Draw GN parallel to the line of impact(not necessarily in the plane 4OB). Draw aGb in the plane AGN, making NGa = NGA N(x)Uy)八(z) and Ga GA and Gb= GB; then by Prop I Ga and Gb will be But the directions of the coordinates are perfectly arbitrary, and the velocities relative to G; and compounding these with OG, we herefore this number must depend on the distance from the origin have Oa and Ob for the true velocities after impact. lone, that is By Prop. II. all directions the line aGb are equally fA()f(e)=p(x2+y2 +22) It appears therefore that the velocity after impact is comp the velocity of the centre of gravity, and of a velocity equ m Solving this functional equation, we find velocity of the sphere relative to the centre of gravity, which may with equal probability be in any direction whatever If a great many equal spherical particles were in motion in a If we make A positive, the number of particles will increase with perfectly elastic vessel, collisions would take place among the the velocity, and we should find the whole number of particles particles, and their velocities would be altered at every collision; so infinite. We therefore make A negative and equal to -1 a2, so that that after a certain time the vis viDa will be divided among the the number between x andx dy particles according to some regular law, the average number of NCe-( /an) dx particles whose velocity lies between certain limits being ascertain able, though the velocity of each particle changes at every collision Integrating from x =-ao to x =+ oo, we find the whole number Prop. iV. To f nd the average number of particles whose velocities of particles tween given limits, after a great number of collisions among a number of equal particles N,∴C