158 Mr, Ivory on a new Method of deducing 
The arc CC' is the measure of the angle at the sun's centre, 
contained by r and r' ; and it is known by means of the true 
anomalies, since it is = u + 5 ot'j by means of the formula, 
cos. CC'= PC and PC' are the complements of the he- 
liocentric latitudes; and KK' is the difference of the heliocentric 
longitudes. Now in the spherical triangle CPC', we have 
sin. CC': sin. P, or sin. KK' : : sin. PC', or cos. sin. C; 
and in the right-angled triangle KCN, of which the angle N is 
the inclination to the ecliptic, denoted by I ; we have 
Rad. ; cos. KC, or cos. / : : sin. C : cos. I ; 
therefore by combining these two proportions, we get 
T sin. KK' ; 
cos. 1 = -r — . cos. / COS. I . 
sin. CC 
The resolution of the right-angled spherical triangle KCN, 
will give the longitude of the ascending node, and the place 
of the perihelion on the orbit. For the side KN is the differ- 
ence in longitude between the comet at the first observation 
and the ascending node: and the hypothenuse NC is the an- 
gular distance on the orbit between the same node and the 
comet at the first observation ; and from this it is easv to find 
the angular distance between the node and the perihelion, 
which will fix the place of the perihelion on the orbit. 
11. In applying the preceding method, the examples in 
Legendre's Memoir and the supplement to it, have been pur- 
posely taken : first, because these make a selection of instances 
greatly varied in their circumstances : secondly, because the 
results obtained by the formulas investigated in this paper can 
thus be compared with his, and other methods, and with the 
corrected elements, which are likewise given in his Memoir. 
