172 DOOLITTLE—ORBIT OF DOUBLE STAR Y 518.  [April3, 
second Law, which states that the areas swept over by the radius- 
vector are proportional to the corresponding times, will evidently 
be true, provided that in the apparent orbit these radii-vectores are 
drawn from the principal star instead of from the focus. 
Having plotted the series of measures as above described, the 
first step in the determination of a double star orbit is to draw the 
apparent ellipse in such a manner that it shall represent them rea- 
sonably well; the various sectorial areas are then measured with a 
planimeter, or otherwise, and the trial ellipse changed in shape and 
position until finally, after several trials, the measured positions and 
the law of areas are both approximately satisfied. 
To fix the shape of the true orbit and its position in space, and 
to predict the future motion, there must next be determined the 
following seven elements : 
(1) Zhe Period, P. This can be measured directly from the 
apparent ellipse, since, by Kepler’s Law, any secturial area is to 
that of the whole ellipse as the time occupied in the description of 
the area is to the Period. 
(2) Zhe Time of Periastron Passage, T. This is the date at 
which the companion passes the nearer vertex of the true ellipse. 
It can evidently be found from the apparent ellipse by an applica- 
tion of Kepler’s Law. 
(3) Zhe Eccentricity, e. This, since it is a ratio, can be ob- 
tained from the apparent ellipse. 
(4) Zhe Inchnation, ¢, of the true orbit to the tangent plane. 
(5) Zhe Longitude, 2, of the intersection of two planes. 
(6) Zhe Longitude, 4, of periastron. 
The last three elements are obtained ‘by solving a spherical tri- 
angle. The longitudes are measured from the hour circle passing 
through the star, from the north point in the direction of motion. 
(7.) Zhe Semi-Major Axis, a, 
The elements of the true orbit as thus obtained enable us to 
predict the direction and distance of the companion for any time. 
The next step of the computation is to obtain the computed distance 
and direction at the date of each observation. A comparison of 
the computed with the observed positions furnishes a basis for im- 
proving the elements by the principles of Least Squares. The 
same process is repeated with the improved elements, until a satis- 
factory agreement between the computed and observed positions is 
obtained. 
