28 



C. V. L. Charlier 



and 



,„ . mJu.^-^ mJ uJ 4- 2m, nu ii, 



ni^ «1 ^ + ^<2 ^ = — — — ~ — — ^— i— ^ 



But 



so that 



w,^ = wîj [m^ A- ni^ — m^) — {m^ + m^) — 

 = Wig [m^ -\ — wij = w?2 [niy + — 



111 j ^ 111 j . _j_ ^ " 1' 



Putting 



T ==^m, {t,,' + + tv,' )+im, {u,' + v,' + f.;2^ ) 



we deduce 



y _ 2" m2 _ 2 



2 + JW-g) 



where as usual to denotes the relative velocity. 

 This is the theorem of Carnot. 



It shows that the Jcinetic energy is always diminished at a collision. 



The combination of these theorems regarding the integrals of the areas and 

 the energy integral leads to important cosmogonie conclusions, which were first stu- 

 died by the late sir Robert Ball, I have no opportunity here to enter further 

 into this subject. 



Regarding passages we found that the relative velocity w' after the passage (at 

 infinite distance) was equal to the relative velocity co before passage. At collisions this 

 theorem is evidently not valid, as the relative velocity after the passage is always = 0. 



22. The number of stars »thrown out» at a collision. 



Repeating the same reasoning as in § 4 we find for the number of stars 

 which, owing to collisions, are thrown out from the dominion At, in the time At 

 the expression 



(42) {o\li) = AilAz,AtjA^d<!^dbh^.<yf,f^, 



where, however, the integration is to be performed between the following limits : 



■Mg, î^2' ^^^2 between — co and + 00 , 

 4* * U » 27r, 



I » 0 » 



h. e. the same limits regarding ']> and the velocities, but different limits regarding h. 

 The integration may be performed regarding the variables <^ and h and we get : 



(42*) (out) = AÜ As, At n d^ j du^ dv^ dw^ fxf^^- 



