378' Astronomical and Nautical Collections. 



§. 75. 

 In order to have a clear view of the whole proceedinc, we 

 may represent the three hypotheses in this manner : 



1 Hyp. 2 Hyp. 3 Hyp. 



Longitude of the $2 Si Q, + p fi 



Inclination t i i + q 



p and q amounting to 10, 15, 20, or even more minutes. For 

 each of these hypotheses, and for three observations, we calcu- 

 late, according to §. 71, |', |", |"', /, r", and r'" ; and then, by 

 ^ 74, li and k". We next find the times in each hypothesis, 

 between the first and second, and between the first and third 

 observations, calling them, 1st r and t", 2ndly t" + I, t" + o, 

 3rdly t' + m, t" -\- s; while the observed intervals are t' and t". 

 If now the true longitude of the node be JB + x, and the true 

 inclination of the orbit i + y, we have the equations 



xl . ym , I xo , ys ^„ „ , , 



— + - — — r— T, — 4- 2—=f—r ; and hence a'rr 



p q p q 



(t' - r') sp - jt" - r") mp ^^^^ _ (J - r')oq-{t" - r") Iq . 



mo — si ' mo — si 



giving us the true longitude of the node, and the true inclination 

 of the orbit. The true values of r", r"\ |', and |"', are then 

 found by interpolation ; taking for any magnitude which may 

 have been found in the three hypotheses B, B +/, and B + </, 



the true value B + -I^ + ?M.. It is obvious that, in order to 



P '1 



obtain all possible accuracy, we must renew the operation with 

 three new hypotheses for the longitude of the node and the in- 

 clination, less remote from each other, if x and y should be 

 found materially greater than p and q, or if large values of p 

 and q had been employed, such as 50, 60, or more minutes. 

 For the method is only so far correct, as the alterations of all 

 the other magnitudes concerned may be considered as propor- 

 tional to those of the longitude of the node, and of the inclina- 

 tion of the orbit, which is only admissible for small values of 

 p and q; a limitation which is equally applicable to the method 

 of Laplace, and to that which remains to be explained. 



