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parallax of the fixed stars. 
correspond to values of tt between o and 90°, the np to values 
between 90° and 180° and so on. It is evident, then, that 
when the large star is in the point of its apparent orbit immedi- 
ately in the line joining the mean places of both, its elongation, 
or the value of 6 corresponding to this situation will be 7 r — <7. 
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 parallactic 
ellipse may be supposed small in comparison with the dis- 
tance 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 parallax has the greatest effects in opposite senses on 
the angle of position, will nearly meet it in the two extremi- 
ties of a diameter conjugate to that passing through the 
small star. 
To determine the direction of the conjugate diameter, we 
must have recourse to the general equations of the ellipse 
from its centre with polar co-ordinates. Thus r being the 
length of any semi-dia- 
meter CP, whose angle 
of elongation from CA, 
is ACP = (p and r' , that 
of its semiconjugate CD, 
the angle ACD being 
called <p', we have, H and S being the foci, and calling CN the 
abscissa from the centre, x, by a property of the conic sections, 
SP = a — e x~a — er cos. q; YiY — a-\-ex — a-\-er. cos.cp; 
whence, S P . H P = C D 2 = r' 2 = cd — e 9 r 9 . cos. of. 
But we have also, 
a x (1 — e a ) 
1 — e* . cos. 
r /2 __ a a (i— e 2 ) 
1 — e a .cos . <p' a " 
.2 
