VOL. LXXVII.] 



PHILOSOPHICAL TRANSACTIONS. 



179 



When two heliocentric longitudes, and the cor- 

 responding northern and southern latitude are given, 

 the distance of the node from one of the longitudes or 

 places may be found. Let de be the ecliptic, ap the orbit 

 of the planet, n the node, de the difference between 

 the two observed heliocentric longitudes = a, ep the southern latitude = |3, ad 

 the northern latitude = b, ne the distance of the node from the heliocentric place 

 at E, and corresponding to the southern latitude = x. In the spherical triangles 



adn and pen, 

 in the equation 



sin. (a — x) 



tang. b 

 sin. a. cos, x 



sin. X 



■ cot. n = . 



tang. /3 



sin. X . cos, a sin. x 



' tang. 13 



tang, b 



sin. a . tang. /3 



By placing the value of sin. (a—x) 

 By resolving this equation 



sm. X . 



= tanar. cc = :■ -, ~ . 



cos, X ° tang, b + cos. a . tang. /3 



If a, b, and j3, be very small arcs, which commonly is the case with the planets, 

 then sin. a =z a, tang. (3 = (3, tang, b = b, and cos. a = 1 . Hence the spheri- 



cal formula will be transformed into another x = 



a(i 



This formula belongs 



to plane geometry, and may besides be thus demonstrated, dn : ne = ad : ep. 

 •A- Hence dn + ne : ne = ad + ep : ef ; and ne = 



>r 



E 



DE X EF 



D 



. If the difference of the lonaritudes do not exceed 



AD + KF ° 



1°, and the latitudes not greater than 10', the spherical 

 *and the rectilineal formula will agree to a very few seconds. 

 Small faults in the longitude will not very much alter the true place of the node; 

 but very small errors in the latitude are of great consequence. Let the error in 

 ^A the southern heliocentric latitude be fg = -{- d. The 



error in the northern latitude ah = — d. Hence dh : 



Dn = ge : ew, and ew = -y--^-. By subtracting en 

 G^=: r^i the error in the heliocentric longitude of the 

 AA 2 



