130 Prof. Encke on Olbers's Method of 



ner of proceeding of the other methods as well as of Olbers's. 

 The method, as far as it consists not so much in the single for- 

 mulae as in the manner of proceeding to find from the ratio of 

 the distances the distances themselves by means of Lambert's 

 problem, is not changed by this case of exception, but the first 

 approximation becomes indeed less accurate, and the calcula- 

 tion becomes somewhat longer. 



In this case the combination of the two last equations of (13) 

 cannot be applied, and it will be necessary to make use of the 

 two first alone, or in conjunction with the fourth. If we 

 choose first the second, in which p' is already wanting, we may 

 write it thus, 



„ = [rV] sin («' - «) 

 ? [r /] ' sin (a" — a!) ' t 



[rV] Rsin(e-a , )-[r7- f nR / sin(6 f -a f ) + [rr / ] R' / sin(e"-a r ) 

 [r r'~\ sin (a" — ex!) 



The factors [r 1 r"], [r /•"], [r r'] contained in the numera- 

 tor of the latter expression may be considered as the coordi- 

 nates of the sun in reference to a line of abscissae, the direc- 

 tion of which is given by a', and may therefore be designated 

 , by Y, Y', Y". We have next, accurately to quantities of the 

 second order inclusive, 



[^^^[R^] f /J 1AT 



irr'] f [RR 1 ] V f {T T } \ r' 3 RVj 



frH? ~ [rr,/] ii«i & - ^ (X - ±\\ 



[rr>\ ~ [RR'] l_ * [T ; Vr' 3 RVJ 



and 



frr'] [RR 1 ] 

 [rr'J ~~ [RR'] ' 



By the substitution of this value in the latter part, there will 

 appear in the numerator an expression which may be written 

 thus : 



[R' R"] Y - [R R"] Y' + [R R'] Y« 



Now this expression, in consequence of the earth's orbit 

 being a plane, is by (1 2) = 0. There will remain conse- 

 quently only the terms multiplied by (— 3 — p 3 )- If we sub- 



... c [R'R"] [RR"] . . 



stitute in these terms ior- L R - R( J , ;„—, their respective 



T t' 



approximate values — , — , and introduce Y! instead of Y, 



