The Vortex Ring Theory of Gases, 



31 



Again, dxdydz=dx f dy'dz' 



dx x dij Y dz Y = dx\dij\dz\ 



and none of the quantities are functions of 0. 



We have also since the total kinetic energy is not changed by the 

 collision T + T T = T' + T/, and neither of these quantities is a function 

 of 0. Since this is so, we see that the expressions we have assumed 

 will represent a steady distribution if — 



[ 2 r 1 d(p{d^d v d^dcod^ 1 drj 1 d^ l d(v 1 }=y ^(dtj'drj'dgdto'dg'^Tj'^'jduo'J. 



Let us suppose that the collisions are not violent enough to make 

 the vortex rings deviate greatly from their circular forms, and let us 

 consider the effect produced on an A molecule by collision with a B 

 molecule. Let Q' be the potential due to the B molecule, then just 

 as before we have — 



r- 



H 



~dJ^ 



£' = £+(l_i )J +C0C?2Q ^-{fr[ W2Q dt + J*™ fPQ dt + A +C ° d *°dt 

 2 }-<x>dh 2 I J-aodx* J-x>dxdy )-^dxdz 



with similar expressions for ?/ and Here h is drawn along the 

 normal to the A molecule, and the coordinates are supposed to be 

 changed by the collision by only a small fraction of their values. 

 Now the only thing that makes any difference between this case 



f +00 eZ 2 Q 



and the former one is that now dt is a function of v, and 



therefore of iv. If therefore we assume that 3(j? — 2) = q, we have 



dg r dr]'d£'du) f =dtjdr]dgdiv{l+jjqiv— ^-^-dt X 



divj -co dlnr" J 



Now — -dt is proportional to the change in w, and therefore, 

 J -oo dh 2 



by § 29 of my " Treatise on Vortex Motion " is of the form /sin 30, 

 where / is a function of w but not of 0, thus : — 



Similarly 



dg\dr]\d£ f 1 di»\=di; l dri l dg 1 div l {l+±qiv 1 ^ s i n 30}, 



d£ dt] d'C^du) =dgdr)d£div{l -\-\q_io-L sin 30}. 



dvo 



