476] 
ORBIT OF A PLANET FROM THREE OBSERVATIONS. 
419 
48. Consider in the equations just obtained the axis of x as fixed but H as 
variable; that is, let the orbit-pole Z' describe a great circle about the fixed pole X 
(longitude G, colatitude 90°+i7). We have x', y', 1, proportional to linear functions of 
sin H, cos H; viz., writing for shortness 
X c = — a sin G + b cos G, 
X s = (— a cos G — b sin G) sin N — c cos N, 
Y 0 = (— a cos G — b sin G) cos X + c sin X, 
W c = ( f cos G + g sin G) sin X + h cos X, 
W s = (— f sin G + g cos G), 
we have 
,_X C cos H + X s sin H 
x ~ W c cos H + W g sin H ’ 
Y 0 
J ~ W c cos H + W s sin H' 
49. I write 
W e 1 
T7 =— cos A, 
Y 0 m 
W. 1 • A 
= _ sm A, 
Y 0 m 
X l 
~ = — cos A — cot 8 sin A, 
To m 
X l 
^ = — sin A 4- cot 8 cos A, 
Y 0 m 
equations which determine m, A, l, 8, viz., we have 
W x Y 0 
l = m 
tan A = 
X c cos A + X s sin A 
m = 
cot 8 = 
— X c sin A + X s cos A 
Y 0 
Vlf c 2 + W s * 
1 
W c 2 + Ws 2 
1 
Yo YWI+ WJ 2 
{X C W C + X S W S ), 
(X S W C — X C W S ), 
and we then very easily find 
and thence also 
x’ = l + m cot 8 tan (H — A), 
y’ = m sec (H — A), 
y' 2 — (x‘' — If tan 2 8 = w?; 
viz. the orbit-plane revolving about the fixed axis SX', meets the ray in a series of 
points forming in the orbit-plane a hyperbola having the line SX' for its conjugate 
axis. 
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