1892.] Mr Leahy, On the law of distribution of velocities. 323 



of each system, being s, and u the velocity of a particle of the 

 first system relative to the velocity of a particle of the second 

 system. 



Suppose now that after encounter the velocities of the particles 

 become OP i , op t respectively, so that the components of 0P 1 lie 

 between £ and £ + (*£, V l and rj^drj^ £ and £ + «&£; and f/, 

 77/, £/ have similar meanings as the components of op v Since 

 the velocity of the centre of gravity is unchanged by the impact, 

 the condition that the required change shall take place is that the 

 direction of u shall lie within a cone of solid angle dS making an 

 angle 6 with the line of centres at impact. A collision such that 

 velocities OP, op before collision may become velocities 0P 1} op t 

 after collision may be called a " collision of a given kind," and 

 since, as Mr Burbury has pointed out, all directions of the relative 

 velocity after encounter are equally probable if the molecules 

 behave as hard elastic spheres, the whole number of encounters of 

 the given kind in the interval dt will be 



JO 



F(OP)f(op)d^d V d^.d^d v 'd^.irshidt.^ (2) 



per unit of volume. 



Conceive now that the velocity of every molecule of the system 

 is suddenly reversed in sign, the molecules of the solid boundary 

 of the system, if such exist, being similarly reversed as to the 

 direction of their velocities, the position of every molecule being 

 unaltered. The physical properties of such a medium will of 

 course differ in many respects from those of the original medium, 

 but it will at any rate have this property, that all particles whose 

 velocities in the original medium changed from OP to 0P l in the 

 time dt will in the second (or as we may call it the "reversed") 

 medium change in a second interval dt from — 0P 1 to — OP. 

 Now, since the distribution of particles is perfectly regular in 

 space, the distribution of velocities in the original medium 

 " behind " any class of molecules is the same as the distribution 

 " in front " of the same class. Hence the number which in the 

 reversed medium change their velocity — 0P t for — OP by striking- 

 particles with velocity op t is in the interval dt 



d ST 

 F{- OPJfi- o Pl ) %&& ■ m&nlMU ■ -rrshidt . — . . .(3), 



where u' is the velocity of 0P X measured relatively to op 1 and dS' 

 is the angle of the cone, whose axis makes an angle 6 with the 

 line of centres, within which the direction of the relative velocity 

 u' must lie in order that the collision may be one of the given 

 kind. Since the number which change from velocities OP to 

 velocities 0P 1 in the original medium by collisions of the given 



