380 ILLUSTRATIONS OF THE DYNAMICAL THEORY OF OASES. 



Prop. I. Ga and Gb will be the velocities relative to G ; and compounding 

 these with OG, we have Oa and Ob for the true velocities after impact. 



By Prop. II. all directions of the line aGb are equally probable. It appears 

 therefore that the velocity after impact is compounded of the velocity of the 

 centre of gravity, and of a velocity equal to the 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 perfectly 

 elastic vessel, collisions would take place among the particles, and their velocities 

 would be altered at every collision ; so that after a certain time the vis viva 

 will be divided among the particles according to some regular law, the average 

 number of particles whose velocity lies between certain limits being ascertainable, 

 though the velocity of each particle changes at every collision. 



Prop. IV. To find the average number of particles whose velocities lie 

 between given limits, after a great number of collisions among a great number 

 of equal particles. 



Let N be the whole number of particles. Let x, y, z be the components 

 of the velocity of each particle in three rectangular directions, and let the number 

 of particles for which x lies between x and x + dx, be Nf(x)dx, where f(x) is 

 a function of x to be determined. 



The number of particles for which y lies between y and y + dy will be 

 Nf(y)dy ; and the number for which z lies between z and z + 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 + dx, and also between y and y + dy, and also between z and z + dz, is 



Nf(x)f(y)f(z)dxdydz. 



If we suppose the N particles to start from the origin at the same instant, 

 then this will be the number in the element of volume (dxdydt) after unit of 

 time, and the number referred to unit of volume will be 



