Royal Astronomical Society. 387 



direction nfsp) or negative. Sir John adopts for the unit of angles 

 one degree, and for the unit of time one year. 



A series of radii vectores being thus found, corresponding to 

 certain values of fl, the next step is to form from these in numbers 

 the corresponding values of the rectangular co-ordinates a^=/3.cosfl 

 y=p. sin d. And, assuming that the force of attraction between 

 the two stars follows the law of the inverse square of the distance, 

 and therefore that the curve really described is a curve of the 

 second order, and consequently that the apparent curve is a curve 

 of the second order, we must make these numerical values of x, y, 

 (as d?i yp J^s ya, x^y^, &c.) satisfy the equation o=-\-\-a.x-\-fly + 

 yx^ + Sxy + sy-, an equation containing 5 unknown constants. As 

 the number of equations will generally exceed 5, it will be proper 

 to combine them by the method of least squares; and the only 

 question is, what is the function of x and y which shall be supposed 

 d, priori liable to equal error in all ? Sir John Herschel tacitly as- 

 sumes that the function B=l + ax + lSy-\~yx^ •^- Sxy + sy'^ is the quan- 

 tity which with equal weight throughout is to be made as small as 

 possible, or that 2 (B^) is to be minimum. The equations given by 

 this consideration are easily formed, and then a, /3, y, S, s can be 

 determined. 



From these numbers the numerical values of the more convenient 

 elements of the apparent ellipse may be found, and from them the 

 elements of the real ellipse may be found. The formulae for all 

 these transformations are given at length by Sir John Herschel, and 

 they are less complicated than might at first have been feared. 



Thus far the elements necessary to produce geometrical coinci- 

 dence of the concluded orbit with the observed orbit are alone de- 

 termined. The next operation is to determine those elements which 

 relate to the motion in the concluded orbit. For this purpose, 

 angles being taken from the curve based on the graphical projection, 

 and these angles (which relate to the apparent orbit) being con- 

 verted into angles in the true orbit by the formulae lately found, and 

 thus exhibiting true anomalies on the true ellipse, the excentric 

 anomalies are found at once by the formula u—esinu, and the mean 

 anomalies are found. Then every one of these angles gives an 



equation of the form i ^ i ' 



^ m;,=A.?i — I, 



from the assemblage of which the constants k and / can be found 

 by the method of least squares ; and then we have all that is re- 

 quired to form the mean anomaly for any other time t, and conse- 

 quently (as the elements of the ellipse are known) to form the ex- 

 centric and true anomalies. 



The conversion of a place thus computed in the real orbit into 

 one in the apparent orbit, and the comparison of the distance com- 

 puted on the arbitrary scale with the distance measured with the 

 micrometer, and the inference as to the true value of the units of 

 the arbitrary scale, are steps which require no particular explana- 

 tion. 



Sir John Herschel holds out the hope of following up this expo- 



2 C 2 



