12 



C. V. L. Chavlier 



all values of b between d aud ^ B , 



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In order to obtain this result we have to study the changes in the velocities 

 of two stars at a passage. 



7. Relations between the velocities of two masses before and after 

 a passage. I have begun the treatment of this question in Medd. N:o 69. I refer 

 here to the result of this investigation. 



Let 



; 11^, v^, 



be the masses and the components of the absolute velocities of the bodies and 

 before the passage and let 



m/, y/, w/, 



f I I 



be the absolute velocities of the same bodies after the passage. 



Further we denote by v, w and v\ to' the relative velocities before and 

 after the passage, putting 



V = — v^, 

 to = — w^, 



u 



1 ' 



'1 ' 



w 



1 ■ 



Denoting further by /, m, n the direction cosines of the line of symmetry 

 (perihel hne) in the relative orbit, then w^e have 



2m„ 



U-, = 



(9) 



'"'1 + 



Wy = XVy + 



(9*) 



-)- 



-|- W2 



2m ^ 



^ -)- 



2m 



Z (mZ -f î'Wi + %mi 



in [ul -\- vm -\- ivn 



n (ul -j- vm 4- wn 



I [ul vm -\- wn 



-j- 



- — m (ul + VM -j- 



= ^ — n (ul 4- vm -\- wn 



^ ^ m^ -\- m-^ 



If 2d' is the angle between the asymptotes in the relative orbit, we further have 



[10) (J) cos d- — — (ul -\- vm. -\- ivn) . 



