e 



COMMENTATIO ASTRONOMIG A; 97 



« lo the equinoclial, in the same direction npsj and froni Ihe east, so that ihe nf qua- 

 « drant shall corrcspond lo values of tt belween and 90% the np to values betweeu 

 « 90° and 180° and so on, It is evident, then , that when th« large star is in tb 

 « point of its apparent orhit immediately in the hne joining the raean places of both, its 

 « elonpation, or the value of ö ooiTesponding to Ihis Situation will be tt — is. 



« Since it is only the difference of parallaxes which this method can render sensible , 

 « we may suppose the small star a fixed point , and since the dimensions of the parallac- 

 « tic cllipse may be supposed small in comparison wilh the distance of the small star 

 « two tangents drawn from the latter to the circumference of the former, and which ' 

 « of course, mark the Situation of the large one in its. apparent orbit where parallai 

 « has the greatest effects in opposite senses on the angle of position , .will nearly meel it 

 v in the two extremities of a diameter conjtigate lo that passing through the small star. 



« To determine the direction of the conjugate diameter, we must have recourse to Ihe 

 « general equations of the ellipse from its ccntre with polar coordinates. Thus r bein" 

 « the lenght of any semidiameter GP whose angle of elongation from CA is ACP =r &, 

 «and r', Ihat of its semiconjugale CD, the angle ACD being called <p' , we have, H 

 « and S being the foci and calling CN, ihe abscissa from the centre, a:, by a properly of 

 « the conic seclions , 



SP = a — e^ = « — er cos. <?); HP = a + e^r = a + er cos. <J) 5 

 « whence , 



SP . HP = CD» =: r'» =: a' — er^ cos. $=. 

 « But we have also , 



,= = ^' (»-«^) . ^,, _ «' (i-e» ) 

 i — e» COS. (p» ' 1 — e» cos. <J)'»" 



« Eliminating from these three equations, r and.»-', we gel the relalion required be- 

 « tween «p and (p', tiz. 



. ,„ sin. <*» 



COS. cj)^ = ■ x . 



1 — (2 — e= ) e= COS. (p» 



« This aequation will however be reduced to a much more convenient form for our pre- 



« sent purpose by a Iransformation, m'vl. 



tan. $'» = ^'"- '^' '' _ 1— cos. 0' « _ ( 1 — e' ) cos, fp' 

 COS. <p'» cos. ip'» sia. <p:> 



« that is simply : 



— tan. 0' •= ( I — e= ) vol. 

 « the sign — being prefixed in extracling il'c root , because in the ellipse CP and CD 

 « He in differenl quadrants. New in tlie case before us we have : 



<J) ^ !r ■ — (T and CJB' = ^ 

 N 



