8 PROCEEDINGS OF THE AMERICAN ACADEMY 



tinguished from that caused by the inequality of the masses. The 

 proper motion may be eliminated if the observations are repeated in 

 different parts of the orbit of the binary, since its effect would be 

 always the same, while that due to the inequality of the masses 

 would be continually altering, becoming zero and altering its sign 

 twice during each revolution. If the ratio of the masses could be 

 determined micrometrically as described above, the measures with the 

 spectroscope would determine the component of the proper motion in 

 the direction of the line of sight. The principal use of the measures 

 with the spectroscope would be to determine the true dimensions of 

 the orbit, and consequently the distance of the binary. 



Let Q, denote the position angle of the node of the binary, ^ tlie 

 inclination of the plane of its true to that of its apparent orbit, s the 

 distance, and p the position angle at the time of observation ; let c?s and 

 dp represent the annual changes in these quantities. Let us make a 

 transformation to a system of rectangular co-ordinates in which the axis 

 of X shall coincide with the line of nodes, the axis of Z coincide 

 with the line of sight, and the axis of J' be perpendicular to both of 

 them. Then dz will equal the annual change in the distances of the 

 two components from the observer, or will measure in seconds of arc 

 the same quantity that the spectroscope measures by the difference 

 in velocity of the two components. But 



dz ■=. dy tan i and y = « sin {p — Q,) ', 



hence dz = tan i sin (jo — Q>) ds-\- s tan i cos {p — Q,) dp. 



Substituting the proper numerical values we obtain dz in seconds of 

 arc ; it should be remembered that dp must be expressed in terms of 

 the radius, or 57°. 3 must be taken as the unit. This method may be 

 employed if we have an ephemeris of the star, the inclination of the 

 orbit, and the position angle of the line of nodes. If the elements of 

 the orbit are given without an ephemeris, a different formula must be 

 used. Let p denote the real distance of the components, and u the 

 angle from the node measured in the plane of the orbit. If a system 

 of co-ordinates is employed such that X' lies in the line of nodes, T' 

 perpendicular to it in the plane of the orbit, and Z' in the line of 

 sight, we have 



y' ■=: p sin t<, and dz' = dy' sin { = sin i sin u d p -\- p sin i cos u d u. 



If the orbit is circular, u increases uniformly with the time, and 

 p is constant and equals a ; hence dz' ^=. a sin i cos u d u. If 



in this expression du=z~ , or denotes the fraction of the orbit 



