60 M. C. Celle'rier on the Distribution of 



has for its centre, x for its radius ; the straight lines AF, 

 B D are parallel to the axes. We shall then have 



m' = UF(u,v)xdx, (12) 



the function F of the position of the point denoting: — 



(1) If P is interior to OACB . . F(u, v) = 0; 



(2) If P is in the space ACBEA, . F(u, v)= ^"^""^ 



(3) If P is in the space FED, . . F(u, v)=— ; 



(4) If P is in the space DEAU, . F(u, »)=-; 



(5) If P is in the space FEBV, . F(u, v) = -. 



In fact, for OACB we have u 2 + v 2 < x 2 , which is impossible. 

 For AC BE A, u<x and v<x; for FED, u >x and v >x; for 

 both, x being not comprised between u and v, formula (11) 

 is to be employed, substituting in itz=2x 2 — u 2 —v 2 in the first 

 case (where x>aP) } and z = m 2 + u 2 — 2a; 2 in the second (where 

 x<x'). 



For DEAU, u> x and v<x; for FEBV, v>x and u<x; 

 in both cases x is comprised between u and v } and formula (10) 

 is to be employed. 



The function F(w, v) is continuous in value but not in 

 form. 



Fourth Disposition. 



This differs from the third only in this — that the velocity u, 

 common to the molecules of the first kind, is in all directions, 

 indifferently, like those of the second ; in the same manner, 

 required the number of collisions happening only between 

 molecules of different kinds. 



Let us divide the molecules of the first kind into groups of 

 which the velocities correspond to the various elements <w of 

 the typical sphere. On reducing the first kind to a single 

 group, we shall be brought back to the third disposition, except 



that the number n is to be replaced by — , and the numbers 

 of collisions which were expressed by m, m' y will now be ex- 

 pressed by j — , -t— ; adding up these values for all the groups, 

 \he factors m, m' remain the same, and we have only to add 

 the values of j—, which gives the unit. Consequently formulee 



