THE DISTANCES OF A COMET FROM THE EARTH. 107 



abola, it will be necessary to compute, either the three distances 

 from the sun r, r', and r", or two of these distances, with the (74) 

 included chord or angle. In the former case, use may be made 

 of (77) and (80), or of Table III. ; in the latter, (82), (83), &c., (75) 

 are to be employed. In either case, use may or may not be made 

 of (50), (52), or (54), as circumstances require. 



III. If, in Lambert's equation (vide Explanation to Table II. 

 of Bowditch's Appendix to Mecanique Celeste) for finding the time 

 t" required in a parabola to describe the angle between r and r', 

 when these are given together with the chord c", a quantity q" be 

 substituted, such that V(2— ?") q'i = ;^„ then t"=i^'*(3-3")V^ (^G) 

 and T = ^l'+'^ ^ (3 — q)^q- If from these equations q" and q be 

 found by means of Table IV., then, by using B. [5996] (40), 

 -[TTT = ~' (3-0 (I -cj") and ■^^=-, (3_,,, (i_,/n. Hence it is evident (77) 

 that in the parabola these quantities depend only on the sums " 

 of the radii and the elapsed times. Table III. contains the log- (78) 

 arithms of the quantities ^4trT^ ^^ith the argument :-pr^,.wA (Mem. (79) 

 of Berlin Academy, 1778, p. 148). ^'~^' 



When equations (50), (52), and (54) cannot be employed to 

 test the assumed values of g, the above values of q, r, &c., may 

 be employed in the following equation, which should be satisfied. 

 D representing the perihelion distance, 



4D = (2-?") (r + r')_^-^=(2-2) (r' + r")-^^,. (go) 



This has no other recommendation than that by its use the com- 

 putation of the chord c is avoided. 



Perhaps the following mode of testing the assumed values of 

 g, in which the use of (50), (59), and (80) is avoided, is to be (8I) 

 preferred. It is essentially the same as Olber's method, with the 

 corrections of ypj and of ^ taken into account. 



r'=R'-f5.= — 2Rpcos. 5,. r'''^ R"'+M'g' — 2 Mq R" cos. d^. (gg) 



