2G0 Prof. F. Y. Edgeworth on the Application of 



integral of / and of /(M(U 2 + V 2 )+m(w 2 + v 2 )) should be 

 constant, and the two additional conditions that the integrals 

 of /(MU-fraw) and that of /(MV + wi«) should each be 

 constant. Varying f] as before, we find for the required 

 function, 



/= Const. exp.-A(M(U 2 + Y 2 ) +m(u 2 + v 2 )) 



fA(MU+.fiM£) + Z.(MV + »it7); . (21) 



where the constants are to be determined from the data by 

 familiar considerations. 



III. The third argument purports to show that the loss to 

 the contents of any class (U, V) through collisions with 

 molecules of class (u, v) is repaired by collisions between 

 molecules of certain other classes — provided that the dis- 

 tributions of velocities, say F(U, V) and f(u,v), are normal. 

 Following the analogy of the simpler case, consider the 

 species defined by an m molecule of class u occurring in 

 such neighbourhood to an M molecule of class U that a 

 collision will result within the time At. And let the species 

 now be divided into varieties according to the relative 

 orientation of the molecules at or on the eve of a collision, 

 defined, say, by the angle 6 which a line joining the point of 

 contact makes with a horizontal axis. 



Fig. 1. 



By parity of reasoning it may shown that the distribution 

 will'be stable if F(U, Y)f(u, v) = ¥(U',Y')f(u', v) ; or putting 

 as before <E> for log F, (j> for log/, 



$(U,V) + f M ,r)=M(U' 2 +V' 2 )+m( w ' 2 + V ' 2 ), . (22) 



subject to the condition 



M(IP + Y 2 ) + m(i< 2 + v 2 ) = M(U' 2 + V' 2 ) + m(*// 2 + v' 2 ). 



