26 



Prof. J. J. Thomson. 



resultant of V, and the velocity whose components are dQ/dx, dQ/dy, 

 dQ/dz. 



Let g=laP, rj = maP, £=naP, ta=Yi. 



Then we shall prove that it is possible to determine p and q so that 

 the number of molecules which have the values of x, y, z, £, rj, £, w, 

 between x, y, z, £, 77, w, and x-\-dx, y + dy, z + dz, g + dg, rj-\-dij, 

 £ + d%, no + cZto, and for which the kinetic energy of the molecule and 

 the surrounding fluid is T, is when the gas is in a uniform and steady 

 state — 



Ce~ hT dxdydzdl;di)dgdtv, 



where C is some constant determined by the number of molecules in 

 the gas. 



We shall first prove that this represents a possible distribution 

 among the molecules of the quantities denoted by g,rj, £*, iv, when the 

 vortex rings are moving in a fluid whose velocity varies from point to 

 point ; we disregard for the present the effects of any collisions which 

 may take place among the vortex rings themselves. In this case the 

 rings are supposed to be so far apart that they do not influence each 

 other, so that the velocity of any ring is the same as if the others did 

 not exist. T represents the kinetic energy due to the ring and the 

 distribution of velocity potential O on this supposition. 



We have to prove that if the distribution be represented by this 

 expression at any time, it will continue to be represented by it. This 

 will be the case if the expression 



Ge-Wdxdydzdgdydgdu), 



remains constant as the molecules move about. Now T, the kinetic 

 energy, remains constant, so that we have to prove that 



dxdydzdgdrjdgdiv 



also remains constant. 



Since Q is the part of the velocity potential which is not due to 

 the rings themselves, by the equations on pages 65 and 66 of my 

 " Treatise on the Motion of Vortex Rings," we have — 



da_ ± d*Q 

 dt~ 



dl_ (PQ_dH2_ 

 dt dh 2 dhdx 



dm_ <PQ d%Q 

 ltt~- m dh?~dhdy' 



dn_ d 2 Q_d 2 Q . 

 dt~ n lM~dhdz' 



