420 
ON THE DETERMINATION OF THE 
[476 
50. As already remarked {ante, No. 11), this hyperbola is nothing else than the 
intersection of the orbit-plane regarded as fixed, by the hyperboloid generated by the 
rotation of the ray about the axis SX'. And we thus see the interpretation of the 
constants, viz. 
I is the distance from S along the axis SX' of the “arm,” or shortest distance 
of SX' and the ray. 
to is the length of this arm. 
8 is the inclination of the ray to the axis SX'; 
and for the remaining quantity A, imagine parallel to the ray a line through S 
meeting the sphere in L {L is the pole of the separator), I say that A — H is the 
angle LX'Z': or (what is the same thing) drawing X'L to meet IIZ'Y' in A, we have 
IIA = A = H + ZA, or (what is the same thing) Z'A = A — H. 
51. To verify this, observe that the cosine distances of L from X, Y, Z, are as 
f : g : h; and thence its cosine distances from X', Y', Z', are as (f, g, h$a, /3, 7): 
(f, g, h$V, ft', 7') : (f, g, h]£a", ¡3", 7"); say, for a moment, as f' : g' : h'. 
Now LA is the perpendicular from L on the side Y'Z’ of the quadrantal spherical 
triangle LY'Z', and we thence have 
h' cosAF' . . , /A rr . 
—. = ini? — f an A/ = tan (A — H), 
g cos AZ V /> 
if A has the geometrical signification just assigned to it. But this equation is 
g' cos {H — A) + h' sin {H — A) = 0, 
that is 
. s' cos H + h' sin H 
tan A = —— JT —T-y Tt , 
— g sm H + h cos H 
or substituting for g', h' their values, the numerator is 
f {a! cos H + a" sin H) + g (/3' cos H + /3" sin H) + h (y cos H + y" sin H), 
which is 
and the denominator is 
= — f sin G + g cos G, = W s , 
f (— a! sin H + cl" cos H ) -h g (— /3' sin H + ß" cos H) + h (— y sin H + y" cos H), 
which is 
= (f cos G + g sin G) sin N + h cos N, = W c , 
so that the formula becomes 
W 
tan A = F ;, 
which is the original expression of tan A.