390 R&yal Astronomical Society. 



the determination of the fifth coefficient, we shall have an equation 

 of condition which must be satisfied, or whose failure will prove 

 that our operations or assumptions are in some part erroneous. 

 When there are more than four equations, all can be used in methods 

 analogous to those which are well understood in other investiga- 

 tions, for correcting the result. 



We must, however, express our opinion that this part of the 

 operation appears the most obscure, as well as the most delicate 

 and difficult, of the whole. 



M. Villarceau remarks that the final determination of elements 

 will in all cases require observations separated by a considerable in- 

 terval from the rest, 



M. Villarceau has lately communicated to the Academic another 

 method. 



The following are the principal results in the two cases which 

 M. Yvon Villarceau has specially examined : — 



In the instance of ( Herculis, the stars are so unequal that there 

 can be no possibility of confusion between the two. It was seen 

 double in 1782, but there is reason to think that it was seen as 

 only one star between 1795 and 1802, and also between 1828 and 

 1832. M. Struve, expressing himself very doubtful, seemed to 

 suppose that the periodic time might be about 14 years. (See the 

 MensurtB Micrometricce.) A valuable series of observations, how- 

 ever, having been made at Pulkowa, extending to 1847, the whole 

 of which have been communicated to M. Yvon Villarceau, he has 

 deduced from them an orbit in which the excentricity = sin 27° 

 nearly, and the periodic time is 36^ years. The measure of 1782 

 and those from 1826 to 1847 appear to be represented with all 

 desirable exactness. [In comparing the computed and observed 

 angles of position, we are glad to see that M. Villarceau has con- 

 verted their eflFects into expressions measured by seconds of arc] 

 The remarks, too, made by M. Struve about the time of the union 

 of the two stars observed by him correspond exactly to the positions 

 given by M. Villarceau's elements. Those of Sir W. Herschel do 

 not correspond. M. Villarceau suggests that, at a time when the 

 small star really was hidden. Sir W. Herschel may have been mis- 

 led by a false image of the large star ; and that, when the image 

 of the star was deformed, he may have estimated the deformation 

 in the wrong direction. He desires, however, specially to submit 

 these conjectures to the judgement of Sir John Herschel ; and we 

 trust that Sir John Herschel will not decline to undertake the 

 honourable task to which he is invited. 



M. Villarceau concludes with pointing out that this star presents 

 a remarkable illustration of the amount of uncertainty which may 

 rest upon the determination of double-star elements, when based 

 upon a limited series of observations. If we had only to satisfy 

 the observations extending from 1828 to 1847 (or through more 

 than one-half of a revolution), we might have represented them by 

 systems of elements in which the excentricity varies from 0*44 

 to r63, that is, the orbit might have been an ellipse, a parabola, or 

 a hyperbola. 



