Illustration of the Theory of Gases. 431 



so that 



q(v — V) cos 6 — u — u'. 



By the equation of energy 



u?-u' 2 = q(v 2 -Y 2 ): 

 From these we find, as before, 



2tfcos 2 / v \ 



l — qcos 2 0\ cost// 



This may be regarded as a generalization of (6) ; and we 

 see that it may be derived from (6) by writing y/cos 6 for v, 

 and (/ cos 2 6 for g. In applying equation (10) to determine 

 the stationary state, we must remember that the velocity of 

 retreat is now no longer iv, but to cos 6, so that (10) becomes 



— >f(io)dio= I ic cos 6f(iv)dw. 



The entire effect of the obliquity 6 is thus represented by 

 the substitution of v/cos for v, and of q cos 2 6 for ^, and since 

 these leave qv 2 unaltered, the stationary state, determined by 

 (15), is the same as if = 0. 



The results that we have obtained depend entirely upon the 

 assumption that the individual projectiles are fired at random, 

 and without distinction between one direction and the other. 

 The significance of this may be illustrated by tracing the 

 effect of a restriction. If we suppose that the projectiles are 

 despatched in pairs of closely following components, we should 

 expect that the effect would be the same as of a doubling of 

 the mass. If, again, the components of a pair were so pro- 

 jected as to strike almost at the same time upon opposite sides, 

 while yet the direction of the first was at random, we should 

 expect the whole effect to become evanescent. These antici- 

 pations are confirmed by calculation. 



By (5) the velocity vj, which on collision becomes u, is 



, 1 + q - 2q 



l—q l—q 



so that the velocity, which after two consecutive collisions 

 upon the same side becomes w, is given by 



, 1 + q - %q 



_ l + 2g+(/) - ±qv 



-l-2q+(q-Y+l-2q + (q 2 y 



