476] 
ORBIT OF A PLANET FROM THREE OBSERVATIONS. 
421 
52. We might in the equations 
x : y' : 1 = (a, b, cja', /3', y) : - (a, b, c#>, /3, y) : (f, g, h$a", fi", y") 
consider for instance G or N as alone variable, and then eliminate the variable 
parameter so as to obtain a locus; but the results would be complicated and the 
geometrical interpretations not very obvious. 
53. I assume (as was done before) N = 0, G=b — 90°, H = c, that is, the position 
of the orbit-pole Z' is longitude b, colatitude c, and the axis SX' is the line of nodes 
or intersection of the orbit-plane with the ecliptic, viz., the longitude of this line is 
= b - 90°. 
The formulae become 
or if these are 
x' : y' : 1 = (a cos b + b sin b) cos c — c sin c 
: — a sin b + b cos b 
: (f cos b + g sin b) sin c + h cos c, 
, _ X c cos c + X s sin c 
x ~ W c cos c + W s sin c ’ 
, Y 0 
J W c cos c + W s sin c ’ 
the values now are 
X c — a cos b + b sin b, 
X s = c, 
Y 0 = — a sin b + b cos b, 
W c = h, 
W s = f cos 6 + g sin b, 
and thence forming as before the values of tan A, l, m, cot S, and putting for shortness 
V Wc + Wg, = Vh 2 + (f cos b + g sin b) 2 , = £1 
we find after some easy reductions 
A f J g ■ 7 
tan A = r cos b + r sin b. 
h h 
m = ^ (— a sin b + b cos b), 
l = i [(ah — cf) cos b + (bh — eg) sin 6], 
cot 8 = —^ (— a sin b + b cos b) (— f sin b + g cos b), 
= ^ (— f sin b + g cos b),