10 



c. V. L. Charlier 



How great is the number of stars within this volume? Or, more fully ex- 

 pressed, how many stars with velocities within As^ have coordinates within Aü^? 

 According to our general formula (1) this number is 



Integrating this expression, for all values of 4' between 0 and 2Tt, for values 

 of b between d and ^ B — where d denotes the smallest distance at which passages 

 take place — and further over all values of m.^, v^, we get the total number of 

 passages described by in the time Af 



At each passage a star is thrown out from zlsj Hence the expression (5) 

 gives also the desired number of stars thrown out from Ae^. 



Regarding d it may be observed that it represents the smallest distance at 

 which passages take place. If the distance is smaller than d, we have to do with 

 collisions. It follows that d is approximately equal to the diameter of the stars 

 (somewhat greater when the tidal forces are taken into account). 



The number (5) of stars thrown out may more conveniently be written in the 



form 



It may be observed that the integrations in regard to and b may be directly 

 performed, so that the expression (5*) may also be written: 



This expression could, indeed, have been directly deduced without introducing 

 the auxiliary variables b and '\). They are used here only because they aid us in 

 the discussion of the inverse problem. 



5. Number of stars thrown in. Consider within AÜ two groups of stars 

 having velocities within the elementary- dominions Js^" and Ae^" respectively. If a 

 star of the former group passes a star of the later group at a distance b" ■< 5, in 

 an arbitrary azimut it is thrown out from Ae". The whole number in this manner 

 thrown out from As^" (1. e. thrown out on account of passages of stars belonging 

 to Ab^") is, according to (5*) 



* It follows that one and the same star may be thrown out many times from Ae, without 

 being »thrown in» in the intervals. This inconsequence is compensated by reckoning the number 

 of stars »thrown in» in the same manner. 



/g zl^2 /Isg = <ii Atb Ab A'^ At^ . 



= f/a M At b Ab A^ zJe^ . 

 Multiplying by the number of stars m^, namely 



AAÜAb, 



we get the number of passages described by all the stars within zle^, viz 

 (5) AÜ As^ j f^ (0 At b Ab A'\ Ae^ . 



(5*) 



AilAs^At /' Js, d'\dbbi.<\f\f.,. 



