102 SOME METHODS OF COMPUTING THE RATIO OF 



(20) = sin. (a" — a) ^^' q cos. 5 — sin. («" — «') |^ g' cos. 6'— sin. («"—©) |^ 

 fi COS. + sin. («"— 0') ^' R' COS. ©' — sin. («"— ©") R" cos. ©". 



In which a, 6, &c., represent right ascensions and declinations. 



(30) Since the accuracy of the ratios j, or -^, on which may be made 

 to depend that of the resulting elements, will, under similar con- 

 ditions, be proportional to the sine of the angle which the direc- 

 tion of the comet's apparent path makes with the great circle 

 B B' (11), as may readily be shown; the best position of this 

 circle is that which is nearly perpendicular to the direction of 



(31) motion. This condition is satisfied by putting in (28), 6 and 

 d' = 0, and giving to a, q, &c., their proper significations ; 



(32) sin. (a' — O) cos. 0, &c., will then represent the sines of the 

 perpendiculars from the sun's places, upon the great circle pass- 

 ing through the middle observation, which is nearly perpendicular 



(33) to the direction of the comet's motion, a' — a and a — a" will 

 then comprise the whole amount of the comet's motion in the 

 intervals between the observations, and consequently, by the con- 



(34) ditions, the coefficients of g and g'' are as accurate as the obser- 

 vations will give, and are independent of the direction of the 

 comet's path. 



(35) The same result is obtained by putting in (4), d' or 0" = O, and 

 giving to 6, 0, &c., their significations as just stated (32). 



(36) = !g;;% sin. 6 -i- e" sin. 6" - ^^ B sin. e + ^^ R' sin. e' - R" sin. o". 



(37) = tf^ 9 sin. 6 - ff^] q' sin. 6' -^^^ R sin. O + ^^j R' sin. 0' - R" sin. e". 



[rr'] " [rr'J ^ [rr'j ' [rr'J 



(38) The sum of the last three terms being of the second order in 

 T (13a). 



These may be expressed in terms of right ascension and dec- 



(39) lination as follows: — Let C represent the right ascension of a 

 point in the equator, the great circle from which passes through 



