888 Royal Astronomical Society. 



sition with the details of the application of his method to the star y 

 Virginis. 



As an Appendix to Sir John Herschel's paper, it is proper to add 

 that papers have been received by Sir John Herschel from M. Yvon 

 Villarceau (namely, a note on the double star X, Herculis, dated 1849, 

 February 1, a note on the double star ij Coronse, dated 1849, March 

 30, and a letter dated 1849, April 1, containing an exposition of 

 M. Yvon Villarceau's methods), which have been communicated 

 more or less completely to the Academic des Sciences of France, 

 and which therefore cannot be received in the ordinary way as a 

 communication to this Society. It is, however, the wish, both of 

 Sir John Herschel and of M. Yvon Villarceau, and it appears in 

 every way desirable that their results should be made known to 

 this Society, both as containing instructive expositions of a very 

 elegant general method and very curious applications of it, and also 

 as bearing upon any questions which may arise as to the similarity 

 or priority of the methods of Sir John Herschel and M. Yvon 

 Villarceau. 



Assuming the law of gravitation, and consequently the law of 

 elliptic movement, as applying generally to the relative motion of 

 two stars in a binary system, M. Yvon Villarceau remarks that the 

 projection of this curve upon the spherical sky (or rather upon a 

 plane perpendicular to the visual ray) will be a curve of the second 

 order, whose equation will be, 



Y=-ay'^ + bxy + cx"^ + dy + ex ■\-f=o, 

 the origin of co-ordinates b9ing one star regarded as a fixed centre 

 of attraction of the other. The object of the next process must be, 

 to adopt this general equation to the particular observations from 

 which the orbit is to be deduced : and here it is to be observed 

 that M. Villarceau does not confine himself either to the measured 

 angles of position or to the measured distances, but uses both, for 

 the formation of the numerical values of ,r and y corresponding to 

 every observation. Having these numerical values of rectangular 

 co-ordinates, and paying no respect (for the present) to the inter- 

 vals of time between the observations, the following is the method 

 used to accommodate geometrically the curve of the second order 

 to the observed co-ordinates : — 



The principle assumed is, that the constants a, h, c, &c. shall 

 be so determined that if the resulting curve be drawn, and if from 

 every observed place a normal (usually a very short line) be drawn 

 to the curve, then the sum of the squares of these normals, each 

 multiplied by its proper weight, shall be a minimum. This prin- 

 ciple, it is almost unnecessary to remark, is imperfect, inasmuch as 

 it does not in any way take cognisance of the laws of movement, as 

 connected with time; but it will frequently be doubtful, in a pro- 

 blem of such difficulty, whether it is not best to neglect a condition, 

 even of the most essential kind, for the sake of making the solution 

 more simple. 



Putting h^ D' for (^)'+ (?")*' »»^ P ^^"^ *^^ ^^^S^^ °^ 



