424 
J. H. JEANS ON THE DISTRIBUTION OF MOLECULAR ENERGY. 
will each l^e multiplied by linear functions of the coefficients of terms of degree n in 
y, and tlie whole expression will he multiplied by X where 
X = ^ d]) dll dp dY du u" dx" dif. 
Now XF dP is the number of collisions per unit time which are experienced by all 
the molecules of which the q, s, co-ordinates lie within limits dP. Since tfie 
probability of a collision does not depend on the q, r, s co-ordinates, it follows that 
the total number of collisions per unit time is X |F dP. It follows that X is a 
function of h only, and does not depend on the coefficients which occur in y. 
I]i the second integral M, terms of degree n do not occur at all. 
In the remaining terms of erpiation (xix.) terms occur of degree n in q, /*, s, the 
coefficients l)eing of the same form as those occuriing in L excej^t that the factor X 
does not occur. 
We have thus found as many equations as there are coefficients ; in these 
equations every term is a coefficient multij^lied by a constant. The only solution 
of tills system of equations is that every coefficient vanishes. 
This result depends on the assimq:)tlon that n is greater than two. Hence in tlie 
steady state y will contain no terms of a degree higher than the second in q, r and 
At the same time y can contain no terms which are linear in q, r or s. The intro¬ 
duction of these terms would give a law of distribution such that an Infinite nund)er 
of molecules would have an infinite value for di dc 
We may, therefore, suppose tliat, except for an additive constant, y is a quadratic 
function of q, r and .s, in which only square terms occur. 
The equations between the coefficients of this quadratic expression must he linear, 
since they are the coefficients of the terms of highest degree in y, and hence must 
lead to unique value for these coefficients. 
Thus for the type of molecide wliich we have been considering, there is onlv one 
steady state })ossible on the assumptions we Iiave made. 
It is easily verified that 
y = Q + S + V.(xx.) 
is a solution of equation (xix.). For, with this value for y, 1)^/1)/ and both vanish, 
• 0Y fd cy 
so that L and M both vanish, and we have also ,s, = ?’|. Hence everv term 
of equation (xix.) vanishes separately, and the value of y given by eipiatlon (xx.) 
supplies a solution which, as we liave seen, must he unique. This is the solution of 
Maxwell and Pair.TZAr vxx.''" 
* The possiliility of x hoiiig an infinite series has l)een disregarded in tlie ahovc sections. lYe may, 
however, consider that a series which was divergent for any finite values of tlie variables would lead to an 
impossible latv of distrilmtion, whilst a series v Inch is convergent for all finite valne.s, may be treateil as 
the limit of a finite series in which the number of teims is made infinite. 
