664 Proceedings of the Royal Irish Academy. 



since this ratio is entirely unknown, we see that the inequality of light 

 cannot serve to fix even an inferior limit to the parallax ^ 



I have briefly summarised the results of investigations on this 

 subject as far as I know of them, because they are closely allied to 

 the subject of this paper, and because if an effect so small, even if it 

 could be detected, and so inseparably mixed up with the determina- 

 tion of the elements themselves, that it is impossible to disconnect 

 them ; if such an effect has been looked to at one time by astronomers 

 to afford a clue to the parallax, there is some excuse for drawing 

 attention to another inequality of light which is undoubtedly extremely 

 small ; but if it fails to give us the value of the parallax, the failure 

 is due, not to any inherent impossibility in the problem itself, but 

 to the inability of our instrumental means to measure its effect. 



In the case of a double star whose orbit has been determined, if 

 we knew the distance, we could calculate the actual linear velocity at 

 any moment in miles per second : and inversely, if we knew this velo- 

 city, we should be able to deduce thence its distance. We are able 

 by means of the spectroscope to determine the resolved parts of the 

 velocities of each of the components in the line of sight, the difference 

 of which is the resolved part of the relative velocity in the same direc- 

 tion, which will of course enable us to find the whole velocity, and 

 thence to obtain the distance. I proceed to find the relation connect- 

 ing the parallax and the velocity : — 



If V is the velocity in the orbit at the time under consideration, 

 and if Fis the velocity in the line of sight, then Vis the resolved 

 part of V parallel to the latter line. 



Now if, adopting the usual notation, A denote the angle between the 

 line of nodes and the line of apsides of the orbit, and if y denote the 

 inclination of the orbit to the tangent plane to the sphere of the hea- 

 vens at the point occupied by the primary, and if <^ denote the angle 

 between the tangent to the orbit and the line of apsides, then <^ - A is 

 the angle between the tangent and the line of nodes. Resolving the 

 velocity parallel to and at right angles to the line of nodes, we get 

 V cos (<^ - A), V sin (^ - A). The former of these being at right angles 

 to the line of sight has no effect on the velocity parallel to that line, 



while the latter makes with that line an angle equal to y. We 



have, therefore, the whole velocity parallel to the line of sight 



= V sin (<^ -A) sin y. 



Now v = - , where h is twice the area described in a unit of time, 

 P 

 andj^ is the perpendicular on the tangent from the focus occupied by 



the primary. Also /« = — ^ — , ay, h^ being the semiaxes of the orbit, 

 and P the period in years, and p - -^ , whence v = — ^-^ — . But 



