the Molecular Velocities in Gases. 



59 



(2) But if z>\/(ii 2 — v 2 ), we shall designate by 6 r an acute 

 angle such that we have 



\/(w 2 + i; 2 ) 2 -4uV cos 2 6 f = z; 

 and it will be necessary to integrate only from 6 = 6 r to 

 e = ir — 6 r , between which limits V V ' > z. We thus find 



~F(u,v) — k — [\Zu 2 + v 2 + 2 wo cos W — \/ii 2 + v 2 — 2uv cos ff~\ 



J 2uv L 



The factor in brackets having for its square 



2u 2 H 2u 2 -2 v / 2 + v 2 ) 2 -4u 2 v 2 cos 2 6' or 2{u 2 + v 2 -z), 

 the result will be 



F(«,t;) = 



x /2{u 2 + v 2 -z) 



(11) 



Let us now suppose that the number of collisions is required 

 for which one of the new velocities is comprised between two 

 given numbers x and x + dx differing very little. The differ- 

 ence B will then lie between two numbers z and z+dz corre- 

 sponding to those ; and consequently the number sought will 

 be again the value (7) of mf. If z corresponds to x, the 

 squares of the new velocities being x 2 , iiP + v 2 —^, we shall 

 have to suppose x 2 < u 2 + v 2 ; and, moreover, 



(1) If x is the greater of the two velocities, z=2x^ — u 2 — v 2 ; 



(2) If x is the smaller, z=u 2 + v 2 —2x 2 . 



In both cases it will be necessary to take dz=4:xdx, the 

 sign — found in the second signifying merely that x corre- 

 sponded to z + dz and x + dx to z. 



Calling the second velocity x' , x and x / will both be com- 

 prised between u and v, or both not 

 comprised, according to whether we 

 have z < or >\/(ii 2 — v 2 ) 2 - Indeed 

 in the first case the difference of the 

 squares of the velocities, which was 

 + {u 2 — v 2 ), has become z, or has 

 diminished ; the velocities have 

 therefore come nearer to equality, 

 whilst in the second they have re- 

 ceded from each other. It will 

 therefore be necessary to employ B 

 formula (10) in the first case, for- 

 mula (11) in the second. 



The result can be better expressed 

 by regardiDg u and v as rectangular 

 coordinates of a variable point P, referred to the axes OU, 

 OY, the point being in the angle UOV. The quadrant ACB 



V 



E 



F 



D 



> 



U 



