a first Approximation to the Orbit of a Comet, 157 
Let E and E' denote the two angles of elongation (or the 
differences of the geocentric longitudes of the sun and the 
comet); and C and C', the two angles of commutation (or the 
differences in longitude of the earth and the comet) : then 
sin. C = - . —-7 X sin. E 
r cos, I 
sin. C 

r' 
cos. 
cos. I 
X sin. E'; 
by means of which formulas, the heliocentric longitudes of the 
comet at tlie two extreme observations will be known. 
Let YN/^ represent the ecliptic; K and K', the places of the 
comet in the ecliptic, determined by the heliocentric longitudes 
already computed ; PK and PK', the circles of latitude, and 
KC and K'C^ the heliocentric latitudes already found ; and lastly, 
let NCCVz be the great circle in which the plane of the orbit 
meets the heavens: then although it is possible that the comet 
may pass from C to C', either by describing the small arc 
CC', or the arc which CC' wants of a whole circle ; yet from 
the nature of the case, no ambiguity can hence arise, because 
in finding a first approximation to the orbit of a comet, the 
arc CC', which embraces the whole motion in the orbit in the 
interval between the extreme observations, will never con- 
tain a great number of degrees : it is therefore easy to infer 
whether the comet has increased, or diminished its longitude, 
in passing from C to C' ; in the first case, the motion of the 
comet is direct ; in the second case, it is retrograde. Nor can 
there be any difficulty, or ambiguity, in distinguishing the 
Ascending Node, through which the comet passes from the 
south, to the north, side of the ecliptic. The ascending node 
is marked, in the figure, with N ; the descending node, 
with n. 
