[582
the terms of
583]
197
583.
ON SPHEROIDAL TRIGONOMETRY.
[From the Monthly Notices of the Royal Astronomical Society, vol. xxxvn. (1876—1877),
p. 92.]
The fundamental formula of Spheroidal Trigonometry are those which belong to
a right-angled triangle PSS 0 , where P is the pole, PS, PS 0 arcs of meridian, and
SS 0 a geodesic line cutting the meridian PS at a given angle, and the meridian
PS 0 at right angles. We consider a spherical triangle PSS 0 ,
Sides PS, PS 0 , SS 0 = 7, y 0 , s,
Angles S 0 , S, P = 90 , 6, l,
where 7 is the reduced colatitude of the point S on the spheroid (and thence also
7o the reduced colatitude of S Q ) and 6 the azimuth of the geodesic SS 0 , or angle at
which this cuts the meridian SP; and then if S be the length of the geodesic SS 0
measured as a circular arc, radius = Earth’s equatoreal radius, and L be the angle
SPSo, S, L differ from the corresponding spherical quantities s, l by terms involving
the excentricity of the spheroid, viz. calling this e and writing
e cos 70
Vl — e 2 sin 2 7o ’
then (see Hansen’s “ Geodätische Untersuchungen,” Abh. der K. Sächs. Gesell., t. viii.
(1865) pp. 15 and 23, but using the foregoing notation) we have, to terms of the
sixth order in e,
-rf== (1+ P 2 + W fc*+ Si?)s
vl — er
+ (P 2 + h & + 1M4 №) sin 2s
+ + irk 4 ^®) sin 4s
+ yöW^ sin 6s;
and
L =i--|e 2 sin7o{(l-iA; 2 + ie 2 - && + ie 4 )s
~(j\k 2 + ^t) sin 2s
+ sin 4s|,
which are the formulae in question.