THE DISTANCES OF A COMET FROM THE EARTH. Ill 



simplified by putting m = sin. (C' — a) cot. 5, by which they become 

 of the form sin. (C — «') — m, tan. 6', sin. (C — Q) — m tan. 0, &c. 



To find J, (28) may be employed, using right ascensions and 

 declinations ; and for j, (4) may be used, with declinations alone, 

 since sin. d", the coefficient of g", happens to be very small, and 

 sin. 0, sin. &, &c., are elsewhere used. 



For convenience in reference, the known coefficients in (57) 

 may be denoted, in the order in which they are then placed, by 

 «o, 6o' Co» ^o 5 '^hose in (28), by a^, bi, c„ &c. 



log. flo 8.4337430 sin. 6 9.8535231 log. t" 9.2355814 



log. Jo 0^0830200 sin. 6' 9.7926466 log. z 9.1898239 



log. Co 0.0600656 sin. 5'7!3086315 log. i' 9.5143350 



(io- 1-0623875 sin. 8"88 18844 r^ =0.9989317— 0.0693162 p + 9' 



log. ai 9.7505754 sin. 0' 9". 1528482 r'^ = 0.9932688— 1.0974287 p' + 9' = 



log. J. 9"5773059 sin. 0" 9^2974047 »•" = =0.9883345— 1.2953963 9" +e" = 



log. ci 9"8388654 cos. 8 8.5400370 log. ^ 9.9542425 



log.rf, 9.7487023 cos. i5' 9.7408130 log. iL 0.2787536 



6,-0.4297613 COS. S" 9.8139207 



From the direction of the comet's motion, it follows from (30) 

 that neither (28) nor (4) is the most favorable for determining 

 f and f- ; the terms neglected in (28) in the first approximation 

 have also somewhat larger coefficients than in (44) or (4). The 

 latter give for approximations, log. 7 ^0.112 and log. ^:= 9.742. 



As the geocentric motion of the comet is very large, it is prob- 

 able that its distance from the earth is small, and we may as- 

 sume § = i, which gives, 



First Approximation. 

 log. 9 9.523 r 1.0425 log- JFlhy 9-2518 log. ^, 9.99767 



log. q' 9.265 r' 0.9085 log. -,,;,„ J 9.2983 log. j 9.99710 



\~'2~y 



log. 4." 9.635 r" 0.7843 log. >4W? ^■^'^^^ H- f. 9-98935 



(^) 



•2 



