32 



Prof. J. J. Thomson. 



so that neglecting the squares of small quantities 



dg 'drj 'dg 'div' 'dg\dr)\div\=dt;d7]d£dwdg l d7) i dg 1 div l {l + ±qw^ sin 30 



div 



die 



and therefore 



I 2 <yd(fi{di;'dr]'d£'d(v' 1 dg' l d}]\d£'d(jo}= 2 ^^(^{d^d^d^divd^dijYd^du}-^. 



2 2 



Since we see from the form of 7 that — 

 2 sin 307^0=0. 



Thus the condition for a steady distribution is satisfied, and we 

 therefore conclude that a possible distribution of the values of the 

 coordinates among the molecules of the gas is represented by the 

 expression — 



C e~ hT d^d>] d£dtv dxdy dz, 

 where g=laP, r)=maP, g=naP, 10— vi 



and 3(p-2) = 2. 



Let us consider the case when there is no external disturbance in 

 the fluid containing the vortex rings ; the distribution will be uniform 

 in all parts of the fluid, so that the number of molecules which have 

 the quantities gf, y, £, ca, between £, rj, u and |;-fc?£, rj~\-dq, £ + d£, 

 w + dw is independent of x, y, z, and so by the above formula will be 

 proportional to — 



e-^d^d^d^ 



or if the normals to the planes of the vortex rings point uniformly in 

 all directions the number of molecules which have a between a and 

 a + da, v between v and v + dv is proportional to — 



e~ IlT aZp- 1 vV- l dadv, 



or substituting for q the value 3(p — 2) 



e -Wa?p-^v*p-1dadv. 



Though in the kind of molecule we are considering, a and v may 

 be treated as independent variables, still the limits of v depend upon 

 the value of a. For suppose the molecule to consist of n rings 



