[ 386 ] 

 XLVIII. Proceedings of Learned Societies. 



ROYAL ASTRONOMICAL SOCIETY. 

 [Continued from p. 308.] 

 May 11, f\^ the Determination of the most probable Orbit of a 

 1849. ^-^ Binary Star from the assemblage of a great number 

 of observed Angles of Position. By Sir J. F. W. Herschel, Bart. 

 With some Remarks by the President on a Solution of the same 

 Problem by M. Yvon Villarceau. 



In this paper Sir John Herschel refers generally, for the prin- 

 ciple of his method, to a paper published by him in the fifth volume 

 of the Memoirs of this Society, the paper (we may remark) in 

 which was given an exposition of the principles by which the orbit 

 of a double star was for the first time actually determined. He 

 now states his conviction that the method there expounded is, on 

 the whole, the best that can be employed ; and the object of the 

 present paper is, retaining the original principle (namely, of using 

 only the measured angles of position, and rejecting entirely the 

 measures of distance), and retaining the first step of the original 

 method (namely, of smoothing down the irregularities of the angles 

 as -measured, by laying them down graphically, the angles for 

 abscissae and the corresponding times for ordinates, and then draw- 

 ing a curve by hand through the points so found, and using that 

 curve as the representation of the real relation between the angles 

 and the times, and measuring from it the times corresponding to 

 angles which differ by 5°, or by 10°, or any other convenient dif- 

 ference) ; retaining the original method thus far, to complete the 

 investigation by a process entirely algebraical and arithmetical. 



Supposing the times corresponding to equal intervals of angle 

 to be taken from the curve above mentioned, the next thing required 



/// 

 is __ for every 5° or every 10°, &c. This is to be found by the 



following formula, which requires for application only the finite dif- 

 ferences of t for the equidistant values of 9, 



rffl A9l 1 2 3 J 



The next step is, to infer from this the true apparent distance of 

 the stars, as it ought to be measured by a perfect micrometer or 

 measuring instrument. Now every determination of an orbit of 

 double stars proceeds on the assumption of an attraction between 

 the two components, and this requires the supposition of descrip- 

 tion of areas proportional to the time, both in the orbit really 

 described and in the projection of the orbit which we see. Hence 



we must have p^ = constant = 100 (the unit of the radius vector 



dt 



being for the present arbitrary), and therefore p = \/ \QQ.—^ or= 



— 100 -^, according as -— is positive (that is, d increasing in the 

 M dt 



s/ 



