402 
MR. J. II. JEANS ON THE DISTRIBUTION OF MOLECULAR ENERGY. 
Calculated upon the usual assumption, the number of collisions which occur in a 
volume n of the gas and within a time dt, between pairs of molecules of which the 
velocities lie within a range du do div dzs du' dv' did dzs' surrounding the values 
it, V, u\ zs, u', v', w', zs' , is 
Trco'Y dt 
’ a 
F {c, zz) F (c, cs')du dv div dzs du'dd did dzz\ 
Hence referring to expression (ii.) we see that the total increase in the translational 
velocity of the gas, in time dt, is 
TTUrYdt 
n 
F(c,t;T)F(c',CT')[r'aaV“ + r'/3,{TZ- + +.] 
du dV die drs du dd did dzs'. 
If we reject all terms of a degree liiglier than d In r, tliis expre.ssion becomes 
+ ^'')] dv div drz did dd did dzr' 
.... (V.). 
2t2 
Now the functional form represented by f is unknown, but the part of the above 
integral which contains depends only upon J/(ct) (/ct and this can he seen to be 
proportional to p and to involve k. Let us denote jy'(^) dzz by pi, so that I is a 
function of Ji only ; then the part of (v.) vJiich contains a.i, contains p'l~ multiplied by 
7r(i~dt o .. n ,. i? 7 
r* and a lunction oi h. 
The part of (v.) wliich contains depends on I and also on dzs. If we 
write I / ( zs) zd (7n7=pLF (.SO tliat T Is the mean value of zd taken over 
all the molecules of the gas), then this part of (v.) will be p~I'T multiplied by 
TTCl'^dt , p , PI 
and by a lunction ol h, 
2n 
Hence determining the functions of h from a consideration of dimensions, we find 
dK 
in which a^, are constants ; or in terms of H and K, 
itK 
dt 
= apK^'- + ySpK’HI*.(vli.), 
in wlilch the constants a, yS do not In any way depend upon the law of distribution 
of velocities. 
* Using the values for a, ft given in the footnote on p. 401, "vve can find for a, ft the v.alues 
9 mK- ^ 3m 3 ??iK“ ^ 3m 
In this way we can prove the relation H = K, insteac] of assuming it. 
