Statistical mechanics 



13 



8. Instead of I, m, n we now have to introduce the variables b and used 

 in the preceding paragraphs. 



We first deduce a relation between h and v>. We observe that h is equal to 

 the perpendicular from the focus of the hyperbola against the asymptote and tliat 

 this perpendicular is equal to the transverse axis of the hyperbola (which axis is 

 also generally denoted by the letter h). Consequently we have 



(11) ig& = hla, 



where a half the great axis of the hyperbola. 



The energy integral in the problem of two bodies affords the relation 



r _ 



where 



a a 



so that 



At infinite distance between the bodies we get 



a 



or 



a = ^ , ^ 



CO" 



which value, substituted in (11), gives 



(12) , 



which is the required relation between h and -Q-. 



We have to deduce the relations between ^ and the quantities I, m, n. 



The direction cosines of the line of symmetry in the relative orbit referred to 

 an arbitrary system of coordinates X, Y, Z are, according to our assumptions, 



/, m, n. 



The origin of this system of coordinates may be assumed to coincide with the 

 centre of the hyperbola. 



Referred to another system of coordinates S, H, Z, of which the Z-axis coin- 

 cides with the asymptote before the passage, the direction cosines may have the values 



a, -q, c. 



Then we evidently have 



i = sin d cos 'I , 



(13) 'q = sin d- sin 4", 



Ç — cos , 



where the »azimut» tjj is reckoned from the S-axis. 



