THE DISTANCES OF A COMET FROM THE EAKTH. 113 



with either of which the observed places may be satisfied to within 

 one or two seconds of arc. 



The final values are log. 'y = 0.1 154850 and log. '7 = 9.7449468. 



The limit of error allowable in these ratios will be nearly that (io7a) 

 of the errors of ;;; and "4., which can be derived from the time 

 required by the comet's apparent motion to pass over the probable 

 error of a, d, &c. With the best observations, t, f, and t" will 

 be liable to errors which will frequently affect the values log. 7 

 and log. f in the fifth place of decimals, a consideration which 

 may serve to restrict a useless refinement. 



Explanation of the Tables. 



VI. Table I. contains the logarithmic values of the expression 

 I ^^^Jff^ = G = 1 + i^y sin.^ g-\-, &c., with the argument log. sin." g ; 

 and is used in finding y, as shown in (102), &c. 



It might also be used for finding the time required in an ellipse 

 to describe the angle between r and r', when these are given with 

 the included chord c, and the semiaxis a. For, if sin.''^;y = ''+^^'~° 

 and sin.= i s = "-^^^, then x r= oi (e — sin. t — {x — sin. x))- Gauss, 

 Theor. Mot., p. 120. Which may be solved by means of Table I. 



It may also be used in computing an ephemeris in the following 



way. 



Let x^, <, &c., Ir^r'll, ^;, and v^ — v^, be the values of the 

 heliocentric coordinates x, x", kc. of a comet at •any two epochs 

 separated by the interval of time t',, ; the first and last days of 

 the interval for which the ephemeris is to be computed may be 

 conveniently adopted. For any intermediate time we find firom 

 known relations. 



tI' 



nn «' — - ^— -ITS tan. 6' = T7 COS. a, 0' = r^—xj — 



15 



