14? Prof. Encke on Olbers's Method of 



The three observed geocentric longitudes of\ , „ 



the comet, by J 



The three geocentric latitudes of the comet, by . 8, 8', 8", 



The three longitudes of the sun, by 6, 6', 6", 



The three distances of the sun from the earth, by R, R', R", 



The three times of the observation, by . . . . t, t\ t", 



Hence we have 



x — § cos u — R cos 8 

 y = q sin a — R sin 6 



z = q tan 8 



and the analogous equations for the six other coordinates. 

 Substituting these expressions in (12) we get, 

 [V r"] (<> cos a - R cos 6) - [r r"] (q' cos a! - R' cos 6') 



-f [i! r'] (§" cos a"- R"cos 6") = 

 [r' r"] (q~sin a - R sin 6) - [r r"] (§' sin a' - R' sin 6') 



+ [r r'] (q" sin a" - R" sin S") = 

 [r'r"]qtan$-[rr"]q'ta.n& + [rr'] q" tan 8" = 0. 



In these equations there are five unknown quantities, the 

 three p, p', p", and two ratios between the triangles. Two 

 of them, one q and one ratio, may be eliminated. As every one 

 of these equations separately expresses one and the same con- 

 dition, and as they are besides independent of the assumed po- 

 sition of the axis of abscissa?, it will be permitted to simplify 

 them by changing the position of this axis. The most conve- 

 nient forms are the following, which result from the former by 

 changing the position of the axis of abscissae, in the first of them 

 by 6', and in the second, first by a', and again by 6', the 

 third remaining unchanged. 



[r'r"](§cos(a-e')-Rcos(e-e'))-[r/-"]( ? 'cos(a , -e')-R') 



+ [rr'] (q"cos(u"-e ! )-R"cos(e"-e'))=:0 

 [r'r"] (§ sin (a' -«)+R sin (6 -«'))- [>/•''] R'sin (e'-«'j 

 (13) -[rr'] (g"sin(a"~ a ')~R"sin(e"-a')) = 



[r'r 11 ] ( ? sin(a-e') + Rsin (e'-e))-[> r"] e'sin (*'-6') 



+ [ r .ffJ (^sin(a"-e')-R"sin(6"-.e')) = 

 [r 1 r»] f tan 8 - [r r"] ? ' tan 8' + [r /] f tan 8" = 0. 



equations which likewise might have been obtained by the 

 combination of the two first of the preceding ones. 

 Eliminating ?' from the two last, we obtain 



