Full text: The collected mathematical papers of Arthur Cayley, Sc.D., F.R.S., late sadlerian professor of pure mathematics in the University of Cambridge (Vol. 9)

[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.
	        
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