Astronomical and NaiiHcal CoUecliona. 377 



the longitude of the node and the inclination of the orbit ; but 

 here the first, second, and the numerator of the fraction remain 

 the same for all three hypotheses ; and besides the coefficient of 

 tang I J2 T is the same for two hypotheses. In short Euler re- 

 quires 15 logarithms for each observation, this method only 43; 

 while in the computations of Lexell and Nordmark about 57 

 or 60 are employed. 



§.73. 



The problem having been reduced to the solution of two casc.c 

 of spherical triangles, we might easily introduce differential ex- 

 pressions, instead of the three hypotheses, or we might compute 

 the effect of small alterations in the longitude of the node and 

 the inclination of the orbit on the values of ^ K and r. But I 

 have found by experience that the advantage of this mode of 

 proceeding is not considerable : the quantities themselves may 

 as easily be computed on three hypotheses as the differential 

 formulas, which I am the less disposed to insert here, as they 

 may be investigated with very little trouble. 



§. 74. 



Having then assumed three hypotheses for the longitude of 

 the node, and for the inclination, we compute upon each of 

 them for the three observations J^^ K = |, and r : and then the 

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

 observations from the formula 



K — V([''" ~ r'Y + 4 r r" sin 4 (I" — |')') 

 ^" = V ( \j"' — r'] ' + 4 r' ?■"' sin \ ( |"' — |') = ) 

 We then find from k', k", and r', r", r'", the corresponding 

 times between the first and second, and the first and third ob- 

 servations. By comparing these times with those which have 

 been observed, we obtain the true longitude of the node, and 

 the true inclination of the orbit ; and hence, by an easy interpo- 

 lation, the true values of r', r", f , and |"', by means of which 

 we find the remaining elements. 



