454 
ON THE DETERMINATION OF THE 
[476 
and thence 
: 1 
x 2 , y 2> and oc 3 , y 3 , differ from their former values only by terms in £, 77, which may 
we thus see that the foregoing determination of the orbit for an arbitrary value 
of y Xi writing therein 
would be the same 
2 
orbit for the neighbouring position c = 60° + f, and b = 270° + —¡= 17 of the orbit-pole. 
Writing for greater convenience £ = p cos 77 = p sin ^, the indefinitely small quantity 
p will denote the distance of the orbit-pole from A, and its azimuth measured from 
the meridian will be = yjr. We then have y x = — tan -\Jr, and r x = Vl + y x = + sec yjr, or, 
if to fix the ideas, ^ be considered as < + 90°, then r x = sec : we have thus 
(A = ± e as before) A = f (— V| + sec i/r) ; viz., observing that Vf = 1-527, we obtain 
^=-i(A-l) = -'264 
a/t = 0, 
f = sec“ 1 Vi = ± 49° 6', 
^ = sec- 1 (2 VI - 1) = ± 60° 52', 
= sec“ 1 ( V| + 2) = ± 73° 32', 
^ = ± (90° - e), 
A = 
0 
A= £(Vi-l) = +-264 
1 
= + 00. 
98. These results will have to be further considered in reference to the course 
of the iseccentric curves through the point A. I remark here that, although it 
appears that although for eccentricities less than ‘264, and in particular for the 
eccentricity =0, there are real directions of passage from A to a neighbouring point, 
yet there are not through A any real branches of the corresponding iseccentric curves ; 
viz., A is in regard to these curves, an isolated point with real tangents; that is a 
point in the nature of an evanescent lemniscate. As regards the eccentricity = 0, it 
is obvious that this must be so; viz., there can be no real branch through A. In 
fact, the orbit can only be a circle when the intersection by the orbit-plane of the 
hyperboloid which contains the three rays is also a circle ; that is, the orbit is a circle 
only when the orbit-plane coincides with the plane of the ecliptic.