OF THE KINETIC THEORY TO DENSE GASES. 
5 
substituted for it, the substituted sphere having the same vertical velocity, but not 
quite the same position on average, as the original one had at the instant of its 
collision. 
6. If the collision be direct, i.e., the line of centres coincide with the relative 
velocity, the substituted sphere is advanced in position through a distance equal to c, 
in the direction of the line of centres, and that without any loss of kinetic energy by 
the action of the force f If the collision be not direct, there is an advance, and we 
proceed to calculate its value. Let A be the centre of a sphere which comes out of 
collision with u for vertical component of velocity, x the centre of the sphere in 
collision with it, H the point of contact. Let l denote the vector line of centres xK, 
and cos (ul) the cosine of the angle between l and s. Then ^ c cos (ul) is the 
projection of HA on the vertical, and ^ c cos (ul) is its average value for all the 
collisions in question. 
7. Again, let A' be the centre of a sphere which enters collision with vertical 
velocity u, x that of the sphere colliding with it, H the point of contact, and let 
/' denote the vector A'x'. Then, evidently ^ c cos (ul') — \ c cos (ul). 
There will be as many collisions per unit of volume and time of the one class as of 
the other, and the height of the point of contact H, above the base, is on average the 
same for one class as for the other. Therefore, taking the collisions in pairs, one from 
each class, each pair substitutes A for A' as the sphere with vertical velocity u ; and, 
on average, the substituted sphere A is at the instant of its collision above the 
original sphere A' at the instant of its collision by the distance c cos (ul). We have 
next to show that on average of all collisions of N spheres taking place between 
.9 = 0 and ,9 = els, c cos (ul) = k ds, where k = tt c s p. 
8. Let co be the actual velocity of the sphere A as it issues from collision, so that n 
is the vertical component of co, and cos (iuo) = u/co. Let ip be the velocity of the 
other sphere as it issues from collision with A, and E the angle between co and xJj. 
Also let q be their relative velocity, so that cos (coq) = —- cos — . 
Whatever be the values of co, ip, and E, 
