or the Physical Properties of Gases. 405 



4N _» 2 

 is — ^— v 2 e * 2 8v*; and let us find the number of collisions in 

 s/irot 6 



unit of time between one of these (n*i) and the second system 



of molecules. The number of molecules of the second kind in 



unit of volume whose velocities lie between u and u + 8u is 



4N' -Hi 

 — -= — ire p8u. The proportion of those moving at angles 



n/tt^ 3 1 X toft 



between and + 80 with the direction of motion of m^ is 

 isin0S0. 



The relative velocity of m x with these 



= Vm 2 + v 2 + 2uv cos =r, say ; 

 and the number of collisions in unit of time is . 



4N' -t. _ 



irfr . - u 2 e P . i s in 8u 80=]$ r sin d0, say. 



8. We have now to find the proportion of those that strike 

 with a blow greater than a given quantity (c,-say). Consider 

 a molecule of the second system striking m 1 at an arcual dis- 

 tance (f> from the point of direct impact. The relative velocity 

 of the surfaces before impact is r cos </>, and as the molecules 



are supposed perfectly elastic, the blow will be — 1 2 r cos (ft ; 



and this must be >c. If, therefore, fa be such an angle that 

 2ra 1 ?n 2 r cos fa=(m 1 + m 2 )c, all those molecules that fall within 

 a small circle whose radius is s sin fa will strike with a blow 

 > c. The proportion required of the whole number of impacts 

 is therefore 



TT^sin 2 ^! . 9 , ., /m 1 + m 2 )c\ 2 ., /V\ 2 



2—- = sm 2 fa = 1 — ( -^ — ) =1 — ( - ) , say ; 



its 2 ^ \ 2m 1 m 2 r J \r / 7 J 



and the whole number of collisions required 

 = Nr{l-(^y\sm0 80. 



9. The number of molecules of the second kind with veloci- 

 ties between u and u + c^, and making an angle between and 

 + 89 with direction of m ly is 



9TC' « 2 



sin 080u 2 e~W>8u. 



x/vr/3 3 

 Therefore the number of those with velocity u which have a 



* See Maxwell, Phil. Mag. January 1860. , 



