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IX. 



ON A GEOMETEICAL METHOD OF FINDING THE MOST 

 PEOBABLE APPAKENT OEBIT OF A DOUBLE STAE. 

 By AETHUE a. EAMBAUT, M.A. Plates VIII. and IX. 



[Read Jantjary 21, 1891.] 



Those who have ever attempted to compute the orbit of a double 

 star by the graphical method discovered by Sir John Herschel will, 

 I tliink, be disposed to criticize favourably any attempt to afford 

 an aid in drawing the apparent ellipse of the satellite. 



Once the apparent orbit has been satisfactorily obtained, the 

 construction, by which we thence deduce the elements of the real 

 orbit, whether we follow Sir John Herschel in this step or adopt 

 Thiele's still more elegant method, is so singularly interesting as 

 well for the geometrical principles involved, as for the intrinsic 

 importance of the results obtained, that one cannot but regard with 

 regret the amount of licence allowed to the computer in drawing 

 the apparent ellipse through the observed positions. 



It is not easy to see how the most probable ellipse is to be 

 defined. In the ordinary analytical method of solving the 

 problem the ellipse is expressed by the general equation of the 

 second degree, viz. 



ax^ + 2hxy + %^ + 2gx + 2fy + 1 = 0; 



and by substituting the co-ordinates x, y of each point successively 

 in this equation, we get a number of equations connecting the 

 constants a, h, h, g^f, which are then solved by the ordinary method 

 of least squares. 



The geometrical meaning of the process is, however, obscure. 

 I have, therefore, thought that a method which enables us to 

 determine, if not the most probable, at least a very probable ellipse, 

 cannot fail to be of interest. 



The method depends on Pascal's theorem that the intersections 

 of opposite sides of a hexagon ijiscribed in a conic lie on a right line. 



