322 
Proceedings of the Royal Irish Academy. 
In Xo. 19 the angle of position is inferred directly from observa- 
tion. Tiie other eighteen positions are derived from the cnrve. 
Tangents are next drawn to the curve at the points corresponding 
to the dates of Table III. From the intersection of these tangents 
dt 
with the squares upon the ruled paper, the values of — are readily 
ascertained. The last column of the Table contains the values of 
The quantity /— is proportional to the distance between the 
two stars ; this follows from the law of the description of equal areas 
in equal times, a property which, if true in the original ellipse, is 
also true in the projected ellipse. 
According to the process adopted in this method, the real distances 
are discarded (for the present at least), and the projected ellipse 
is to be constructed from the interpolated angles of position and 
computed distances, that is, from the third and fourth columns of 
Table III. 
From a line Sn, Fig. 2, where 8 is the larger star supposed fixed, 
the points 1, 2, &c., 19, are set off, For example, the angle 7 Sn, 
is 328°, and the distance 81 39-1 millimeters, being the angle and 
distance taken from 'Eo. 7 of Table III. 
If the observations were perfect, and the graphical construction 
correct, we should find, assuming that the law of gravitation holds in 
the binary star, that the points 1 to 19 are all on the circumference 
of an ellipse. S would not be situated in the focus, unless in the 
exceptional case where the plane of the ellipse was normal to the 
visual ray. 
It is manifest that no ellipse could pass directly through all the 
points, and it is, therefore, our duty to construct the ellipse which, 
upon the whole, passes most nearly through and among the system. 
The ellipse found by trial is shown in Plate XXIII. (Science). 
This ellipse passes through 1, 3, 11, 13, extremely close to 2, 9, 10, 
16, 17, 18, tolerably close to 4, 5, 6, 8, while it is at some distance 
from 7, 12, 14, 15, 19. 
The remainder of this portion of the investigation will proceed 
upon the assumption that, if the observations were perfect, the pro- 
jected ellipse would not be different from the ellipse thus found. 
Through Jf draw a tangent to the ellipse, and a diameter through 
C parallel to the tangent, then CA {= a^) and CD h'), are the 
projections of the major axis (^?), and minor axis {h) of the real 
ellipse. 
t being expressed in years, and 0 in degrees. 
