a first Approximation to the Orbit of a Comet, 149 
^ sin. (w— ib) t' 
^ smT(ir-«) • T f 1 o . 
which are very exact, when the intervals between the obser- 
vations are not too great. 
The angle u w'hich enters into the preceding formulas, de- 
pends partly upon the value of the quantity ^ ; and it thus 
becomes impossible to assign the value of that angle when I 
cannot be determined. If we compare I with the qua uity 
formerly denoted by ^ (No. 5), w^e shall get f x 
hence it appears that ^ will be indeterminate when ^is so; and 
in such cases therefore the last formulas wdll fail. Now' the 
cases in which ^ becomes indeterminate have already been 
noticed (No. 5) ; they differ by a real distinction from the 
other cases of the problem, and require a separate discussion, 
if we wish to have clear and precise notions on the subject of 
this research : we shall therefore return to the examination 
of them in the sequel. 
8. On account of the formulas (11), the coordinates of the 
comet at the two extreme observations, will depend only upon 
one unknovvn quantity, namely p : and if we substitute the 
values of the coordinates in the following expressions, viz. r® 
= » V == xx' yy' -f- %%' ; 
there will result these other expressions, w’hich contain no 
unknown quantity but p, viz. 
+ 2R cos. X cos. (e — c) .p + p'* 
= R'" 2R'/3 cos. x' cos. (/ — c') , p (2"’ p^ 
V = RR' cos. [e' — «?) + I R/3 cos. x' cos (^ — c') -|- R' cos. x 
X cos. (^^ — r)| xp-f/3 cos. y . 
cos. y = cos. X cos. x' cos. (c' — c) + sin. x sin. x'. 
