a first Approximation to the Orbit of a Comet. 151 
and by substituting this value in the foregoing expression we 
get, nearly, 
4 (t + t')" }’ 
consequently 
^3 _ 4 (t+tQ* 
a 
which is exact to quantities of the sixth and higher orders. 
Now^'^z^ (x— x')"* -j- (y — yy {z-z'y = x~~\~y''-\-z^ 
— 2 (xx' + yy' +2:2') + -f- y'" + z'"- == r"" + r'" — 2V : 
therefore in this last equation all the quantities concerned de- 
pend upon r% ;•'% and V, which themselves contain only one 
unknown quantity, namely p : thus that equation will serve to 
determine p, or the comet's distance from the earth at the time 
of the first observation, on the finding of which the solution 
of the problem depends. 
9. If we now collect in one view all the formulas that have 
been investigated, we shall have the following method for 
computing the distance of a comet from the earth. 
1st. The symbols a- and t' denoting the intervals in days of 
mean time, between the middle observation and the first and 
second observations respectively ; w e begin with computing 
the arc of the earth's mean motion in the time (r t-') ; this 
arc we shall now denote by 5 , and it will be computed by this 
formula, viz. § = being the circumference of a 
circle whose radius is unit, and 365.23638 the length of the 
sydereal year ; so tnat the logarithm of & will be found by 
