Astronomical and Nautical Collections. 383 



again |" and r" from the parabolic elements which have been 

 obtained for the two remotest observations, and compare these 

 values with those which have been already determined accord- 

 ing to §. 75. If there is any material difference in the values, 

 supposing that p and q have not been assumed too great, that 

 the observations are accurate, and that they are sufficiently re- 

 mote, we may then attempt to find an elliptic orbit which shall 

 agree better than the parabolic. I have not been able in this 

 instance to abridge materially the method of Euler, which he 

 has given in the two works already quoted. Instead of the 

 chords k', k", as soon as we have found |', |", I'", r', r", and r'", 

 we must immediately determine the parameter of the ellipse for 

 each of the three hypotheses, by the formula 



_ sin (I" - I') + sin (I'" - I") - sin (!'" - ^') ^ 

 ~ IhTiT^ ri + sin (!"' - f ) _ si" d"' - ^') 

 r"' r' "t" 



which is much more convenient than that which Euler has given 

 in his Theoria, but is essentially the same with the calculation 

 contained in his Recherches. From the parameter thus found 

 we easily obtain the true anomaly for the first observation, the 

 perihelium distance, the times from the perihelium, and the in- 

 terval between the observations. For this purpose I prefer the 

 formulas of the Theoria to those of the Recherches. By com- 

 paring the computed with the observed intervals, we determine, 

 as for the parabola, the correction of the longitude of the node 

 and of the inclination, and we thus obtain the true value of the 

 elliptic elements by interpolation- 



[Note 6. The elegant theorem here introduced may be thus 

 demonstrated. Calling the distances a, b, and c, the angles 

 formed with the greater axis x, x + y, and x + z, the parameter 

 p, and the eccentricity e, the greater axis being I, we obtain, 

 from the well known property of the ellipsis, a = ^ ^ ^ ^^^ ^ 



. _ p^ and c — - - > whence cos x 



^ - 1 + e cos (x + J/)' 1 + e cos (x + z) 



= PUf: cos {X + y) = £f^, and cos (x+z) = ^ ; but 



cos (x + 2/) = cos .r cos y - sin x sin y, whence sin y = 

 2 D 2 



