Statistical mechanics 



15 



so that ß and a design the polarcoordinates of the Z-axis (the asymptote). The 

 rightangular sphaerical coordinates of the same line have in Medd. N:o 70 been 

 denoted by X, [i, v, so that 



X sin ß cos a, 

 \i = sin ß sin a, 

 V = cos ß . 



Introducing the values N' = 0, N = a — 90" into the direction cosines (15) we get 



cos XS = sin a 

 cos YE = — cos a 

 cos ZE = 0 



cos XH = cos ß cos a 

 (16) cos FH = cos ß sin a 



cos ZU = — sin ß 



cos XTj = sin ß cos a = X 



cos YZ — sin ß sin a = [x 



cos ZZ = cos ß = V 

 The inversion of (14) gives 



Z = ^ cos ZS + 7] cos XH + C cos XZ 

 (16*) m =: ^ cos FE + 7] cos FH -|- C cos FZ 



w = i cos ZE -f- -f] cos ZH + C cos Z7j. 



The nine direction cosines in these formulae — and hence also a and ß — 

 may be expressed through the relative velocities before (or after) the passage. We 

 have indeed, according to Meddel. 70, 



u == — Xoj = — to sin ß cos a , 

 (17) V = — [JLco = — (0 sin ß sin a, 



w = — voi = — M cos ß , 



so that 



cos ß = — *'Vw 



sin ß = }/' -\- v'^Uo,- 



cos a = — u/}/' u'^ -\~ v^, 



sin 7. = — V; y + v 



Consequently we get 



w / = — ê —T===. + ■/] — Qu, 



. (SiU , vw 

 wm = ë r - + '1 r = ^) 



W W = — ■(] 1/ W -\- v — LW 



■q \^u^ -j- — C< 



