258 Prof. F. Y. Edgeworth on the Application of 



" unverified " or, after Boole, " intellectual. " An assump- 

 tion of this kind was tacitly made in the preceding argument 

 when it was taken for granted that the appearance (in co 

 sequence of collisions) of additional couples with velocities 

 TJ' andu' ; v! >\J' } within a distance of (u' — V)At from each 

 other, the slower M molecule behind the quicker one, was 

 tantamount to, or attended with, the appearance of additional 

 couples corresponding in all respects to the above description 

 except that the slower body is now before the quicker one.. 

 The underlying assumption might perhaps be formulated as 

 follows : — (It)) A molecular chaos must present the same 

 appearance, both as to the relative position of molecules and 

 as to their velocities, whether contemplated from above or 

 below. Thus corresponding to couples of the first type just 

 now defined with M behind m, there are (on an average) at 

 any time an equal number of couples in other respects similar 

 but with M before m. 



The preceding arguments are readily extended to the case 

 in which there are several varieties of mass, each pertaining 

 to a large set of cylindrical molecules. 



B. Coming nearer to reality, let us next consider the 

 random movements of disks on a plane. 



I. The simplest case under this head — that in which the 

 molecules are equal not only in volume but also in mass — ha& 

 already served to introduce our first argument to readers of 

 the ' Philosophical Magazine ' *. It has been shown that if 

 a disk with velocities U and Y overtake one with velocities u 

 and v, the line joining the centres at the moment of impact 

 making an angle 6 with the axis of U and u, the new velocities 

 of the former are as follows : 



U'= Usin 2 — Vsin0cos0 + i«cos s + «cos0sin0, 1 



y = — u sin e cos e + v cos 2 e+u cos e sin e + v sin 2 e. I ( 17 ) 



If now with these acquired velocities the said disk collide- 

 with a fresh disk of which the velocities are Ui and vi 9 th& 

 angle made by the centre-line with the axis of U being 

 now $i, we shall have for the new velocity 



tr" = U' sin 2 #!- V sin X cos 6 1 



-j-iii cos 2 #! + •*<! cos 6i sin U . . (18) 

 with a corresponding expression for V" ; where for U' 



* See Phil. Mag., January 1913, p. 106. The " spheres" there con- 

 sidered are here replaced by" disks, and the notation is altered. 



