a first Approximation to the Orbit of a Comet, 137 
Tr 
X' = s'x' + / f 
and if these equations be combined two and two so as to ex- 
terminate we shall get 
X — |— G~x^ ^ <r^ s “I” (ts^ ^ X 
^'y + ) / ( 5 ) 
(t'% (tZ' = ( As <rs ' ) : 
Finally le^t A' be written for (0-'^ + ’■> then, by substituting 
the series that cr, A, s, s' stand for, we shall get 
P (t — t) 
= (r + t') 
&c. (6). 
()]Toi ^ jroS 
3. Employing the same marks as in the case of the coor- 
dinates, to distinguish the known quantities of each observation, 
I shall write, 
« 
c, c°j c'\ for the three geocentric longitudes of a comet : 
X, x°, x'; for the three geocentric latitudes : 
p, p°, p'; for the comet's distances from the earth: 
e, e°, e'\ for the earth's longitudes : 
R, R®, R'; for the earth's distances from the sun : 
by means of which quantities, the coordinates of the comet at 
the first observation will be thus expressed, viz. 
X — R cos. p cos. X cos. c 
y = R sin. ^ + p cos. X sin. c 
z = p sin. X : 
and in like manner the expressions of the coordinates at the 
second and third observations will be formed by placing the 
characteristic marks on the several letters. By substituting 
the values thus obtained in the equations (5), we shall get 
these other equations, viz. 
MDCCCXIV. 
T 
