Rambaut — Orbit of a Double Star. 97 



the line PQ perpendicular to OM, and to take M'P and M'Q each 

 equal to OM. We have, then, if a and b are the semi- axes of the 

 ellipse, a = |(0Q + OP), and & = i (OQ - OP), while their direc- 

 tions are those of the internal and external bisectors of the angle 

 POQ, as shown by the dotted lines in Plate YIII. 



To apply this to the case of the secondary member of a double 

 star, for which a number of places with regard to the primary have 

 been determined, it would, perhaps, be necessary, strictly speaking, 

 in order to determine the most probable orbit, to take every possible 

 combination of these places, five at a time, and to take the centre 

 of mean position of the centres so determined as the most probable 

 centre of the ellipse, and the mean of the directions and magnitudes 

 of the semi-axes so determined as the most probable values of these 

 quantities respectively. 



But this complete treatment of the observations, even in the 

 ease of a small number of places, would entail an enormous and 

 utterly disproportionate amount of labour. The following method 

 will, however, be quite sufficient for our purpose. 



The observations of position- angles having been treated in the 

 usual manner, and the interpolating curve drawn through them, 



dt 

 the value of -t^ should be read from the curves, not at every 5th or 



10th degree of 9, as is the usual custom, but at such intervals that 

 the total number of readings may be some multiple («) of five. 



dt 

 With these 5n values of -rj-., and the corresponding values of 9, we 

 da 



obtain, by means of the equation, 



ldt_ 

 ~''4d9' 



the r and 9 of bn points, which if the observations were free from 

 error, would all lie on the apparent ellipse. 



Having thus 5w points, if we take them in n groups containing 

 5 points each, we shall get n dijfferent determinations of the centre 

 of the ellipse, the centre of mean position of which will be a very 

 probable position of the centre. Each group of five points will 

 also, of course, give a value for the direction, and for the mag- 

 nitudes of the axes as is shown in Plate YIII ; but, perhaps, some 

 who have had experience in the art of computing double-star 

 orbits will prefer to trust to the eye for these, except in cases 



