The Vortex Ring Theory of Gases. 



29 



hence dx' dy' dz du> d£ dr\ dX> r 



=dxdydzdu,d£d v d£ (l + {iq + 3(1 - ip) }^ 



so that if 



2=30-2) 



dx 'dy ' dz 'dw ' d% 'dr\ 'd^' — dxdydzdtvd^drjd^, 



and therefore the distribution represented by the expression 



Ce^dxdydzdvod^drfd^ 



will be permanent if there are no collisions between the molecules. 



We shall now go on to shew that this expression will represent the 

 distribution of coordinates and momenta among the molecules even 

 when collisions take place, at any rate if the collisions are not very 

 violent. 



Let us call the group of molecules which have the quantities x, y, z, 

 17, £ iv between x, y, 2, gf, 97, £*, and x + dx, y + dy, z + dz, % + dg, 



rj -j-drj, £+d£, tv + div the group A. The number of molecules in this 



group is 



C e~ hT dx dydzdoodgdrj d£, 



all the symbols having the same meaning as before. 



Let us consider another group of molecules which have their co- 

 ordinates between x v y v z v g^, rj^ £ 1} u l9 and x l -+-dx 1 , yi + dy±, z 1 + dz l , 



+ Vi + dVn Si + c^x, oo l -\-dw 1 . We shall call this group B. The 

 number of molecules in this group is 



J)e~ KTl dx l dy 1 dz 1 d^ d^d^daj-^. 



We shall suppose that the molecules of the A group come into 

 collision with those of the B group, and that the values of the co-ordi- 

 nates after the collision are denoted by putting dashes to the letters 

 which denoted the corresponding coordinates before the collision. 



In my " Treatise on the Motion of Yortex Rings " it is proved that 

 the effects of a collision depend on, in addition to the quantities 

 already specified, the angle which the line joining the centres of the 

 rings when they are nearest together makes with the shortest distance 

 between the directions of motion of the rings ; let us call this angle 0. 

 is positive for the ring which first passes through the shortest 

 distance between the directions of motion of the ring, negative for the 

 other ring, and it may have any value between — 7r/2 and w/2. 



We may suppose that a collision takes place when the shortest 

 distance between the centres of the two rings is less than some 

 assigned value ; it is not, however, necessary to limit ourselves to any 

 particular way of defining a collision. 



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