128 Prof. Encke on Olbers's Method of 



by the trials by (p), the value of M which is now to be consi- 

 dered as perfectly accurate will be 



(27) M - --jpj M + ^ ^ [R R ,^ ^ . M . 



A final solution of the problem, to calculate r 1 from the 

 values of r, r* and k which have been obtained, these values 

 involving already the determination of the whole orbit, and 

 consequently the value of r', has been given by Bessel in 

 Schumacher's Astronomical Memoirs (Abhandlwigeri) ; it leads 

 to the solution of two cubical equations. If deemed convenient, 

 one may likewise make use of the value of log v given in the 

 table, which is to be deducted from the logarithm of the in- 

 terval in order to obtain that of the area of the triangle. The 

 value of 7 J is thus found by the same interpolation as above. 



There is, however, a case to which the method as hitherto 

 deduced cannot be applied. A perfect understanding of it is 

 most easily obtained by reverting to the original formulae (14-), 

 and by comprising the values of M' and M" in a different man- 

 ner. If we conceive the direction of the sun's longitude for 

 the second observation to be S', and the directions given by 

 the three places of the comet to be C, C, C" and all marked 

 on the sphere of the heavens, and then suppose the triangles 

 between the pole of the ecliptic and S' C, S' O, S' C" drawn ; 

 and if we designate in each of them the sides S' C, S' C, S' C /f 

 by 0-, o-', cr", and the angles at S' by X, X\ 2", we have by sphe- 

 rical trigonometry the following six equations: 

 sin <r sin X = cos 8 sin (a — 6') 

 sin <r cos X = sin 8 

 sin cr' sin 2? = cos 8' sin (a! — 6') 

 sin cr' cos 2 l = sin 8' 

 sin <r" sin 2" m cos 8" sin (a" — 6') 

 sin cr"cos 2 n = sin 8". 

 Deducting the product of the second and third equations 

 from that of the first and fourth, and performing the analo- 

 gous operation with the last four of them, we have 



sin a sin a' sin (2 —2') = cos S cos V { tan V sin (<* — 0') —tan S sin (a'— 0') } 



sin</sin<j"sin(2'— 2")=cos$'cosS"{tan$"sin(>'— 0')— tand'sin(*"— ©')} 



or by substituting these values in M' and M", 



M' - si" * sm (2 — j*) sec 8 



!7 sin cr" sin (2' - W) ' sec 8" 



" sin o-" sin (J5* - X") ' sec 8" * 



