108 SOME METHODS OF COMPUTING THE RATIO OF 



(83) COS. {v" — v)z=^^,[RR" COS. S, — ( R" cos. S^+RM cos. 8,)q-\-M q^ cos. 8^']. 



(84) In which M=t!l, v" — v the angle between r and r", and 5i, 82, 

 &c. the angles comprised between R g, R" g" = K" M g, &c., 

 which are computed as follows : — 



(85) COS. di ^sin. sin. 5 -|- cos. O cos. 5 cos. (© — a), &c. 

 When (31) is used, d = d" = 0, and then 



cos. ^1 = cos. cos. (O — a), &c. 



(86) The value of (1 — q') (76) may be thus expressed: 1 — q' = 

 2Vr-;^cos.H..'-.) ^hich, substituted in t' = <-i±p' (3 — 9') Vq', should 

 give the true value of j'. Ordinarily q' is not ascertained with 

 much accuracy in this way, it being uncertain in the same degree 

 as the value of the chord computed as in the method of Olbers : 

 in the present instance, the angle {v" — v) is adopted instead 

 of the chord, with a view to the correction of the assumed values 



. of ^ and ^^. The angles between r, r', and r" are found by 



(87) trial from the relations i:^^'=^ '(^ and (v"—v') + (v' — v) 

 = (^// — j;) ; or (v' — v) may be found directly from the equation, 

 tan. (v'—v)= , ^'In-^'"""^ . With (v'—v) thus found, r' is com- 



f88) puted from r'=^-^ sinjr"-.) ^"_ ^j^jj jj^ggg values of r, r', and r", 



^ '' -t^ [rr"] sin. (c' — v) 



(89) P^ and ^^ may be corrected by (78), or by (86) and (77), or 

 by Table V. 



(90) IV. The values of j, &c., determined on the supposition of a 

 parabolic orbit, will be affected by the introduction of an eccen- 



(91) tricity by terms of the order ^j a being the semi-axis of the orbit. 

 (9io) In order further to correct ^ and p^, two heliocentric distances 



must be found, with their included angle or chord, by suppos- 



(92) ing (90) to be correct. Thence, by (87) and (88), the third 

 heliocentric distance and the remaining angles. 



(93) When (59) is employed, either of the above angles may be 

 found by the solution of the spherical triangle CEC", &c., in 



