THE DISTANCES OF A COMET FROM THE EARTH. 105 



0= - [sin. (C'-«0 -'^^4^^ta„. 6'] gi ,' cos. ^' - [sin. (C- ©) _ (57) 

 ^-^^'-- «] '^' ^ COS. 0+[sin. (C'_G')-=-^f?^tan. e'] '^ 

 ii' COS. 0' - [sin. ( C - 0") _ ^-^p^ tan. e"] K" cos. ©". 



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

 But the original forms are perhaps to be preferred. It is to be 

 observed that C is not here arbitrary, as C is in (41), (42), (43). 



From these equations it is evident, that, when l!:lr!!l and ^-^^ are 



[ri-lj (rr'I 



known, g, g', or g" may be found ; also, that d, &, and 0", in (58) 

 the coefficients of g, g\ and g", which represent the deviation of the 

 path of the comet from the arc of a great circle, are of the same 

 order with J and J', or of the second order, generally, in r, but 

 affected by the whole amount of the error of the observed places. 

 There is still, however, even when the coefficients of g, g', and g" 

 are =0, a relation to be sustained between i^i and tr!J which r.oi 

 may be made use of m correcting their assumed values (9), (48). 

 Excepting the case when the path of the comet is in the ecliptic ; 

 (50), (52), and (54) then becoming indeterminate. (go) 



By assuming any probable value of g, from (48) may be found (6i) 

 corresponding values of g' or g", from which may be computed (62) 

 the quantities necessary in the equations of III. or IV. for obtain- 

 ing values of l^ and ^j, which, when g has been correctly as- 

 sumed, will satisfy (50) or (52), &c. 



It is here to be observed, that, in determining the five elements 

 of a parabolic orbit, there are given six conditions, dependent on (63) 

 a, a', a", d, d', and d", to determine five unknown quantities ; and 

 therefore either of the quantities a, a', &c., may be rejected. 

 Where the direction of the error in a, a', &c., is unknown, that (64) 

 should be rejected in which it has the greatest influence, which 

 is d, 6', or 0", in (50), (52), and (54); and this is a reason for 



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