1842.] • Excess on a Spheroidal Triangle. 37 



_.. a b sin C . . 



k = ~ o • — tt/> m which the denominator being constant, and the 

 2 r 2 sin 1" 



quantities in the numerator those which occur in the calculation of the 

 triangle, the calculation of the excess becomes as simple as need be. 



To adapt this to the case of a spheroidal triangle, all that is necessa- 

 ry is to substitute for r 2 the expressions for the normal and meridional 

 radius of curvature. Assuming a and /3 to denote the polar and equa- 



torial radii, and f = -L_^_ j an d A the latitude, the expression for 

 a 



the normal is v = — a \ + * 1 — and for the meridional radius 

 (l + a 2 cos 2 A)i 



«(l + * 2 ) a 2 (l-H 2 ) 2 ; 



7 — r s ar-r 3 . hence the product of the two yv = — — i 57- 



(1 + £ cos^A)l ' ' l + £ 2 cos 2 X 



and this substituted for r 2 gives the excess on a speroidal triangle. 



E „ = a b sin C (1 + _e* cos«A)*_ a b sin G ( 1 + t? cos A ) 2 

 2 « s sin 1" (TT?) 2 2 /3 2 «* !" 1 + e 2 



a b sin C ( 1 + ft ™*% 2 _ a 6 sin C 1 + 2 £ * cos*A 

 2 a (3 sin 1" (l + e «)| = 2 a (j sin 1" 1 + | £ 2 



Of the above expressions, the first three are identical and rigorous. 

 The last is an approximation got by actually performing the operations 

 indicated in that preceding it. It is however, sufficiently close for any 

 ordinary purpose, as the quantities omitted affect only the 7th place of 

 decimals. It appears from it that, when 2 cos A = ~> or cos^A = t, 

 that is, in latitude 30°, the excess on the spheroidal triangle coincides 

 with that computed by the mean radius. In lower latitudes the sphe- 

 roidal excess is greater than the spherical, and only in latitudes higher 

 than 30 is it less than on the sphere. 



To render this formula practically useful I have computed in the fol- 



1 1 H" 



lowing Table the value of the logarithm of — or for 



2 y v sin 1" 2 y v 



every degree of latitude. I have assumed the values of the polar and 



equatorial radii of the earth as deduced from the comparison of the 



whole European with the Indian arc, as far as Kahanpur in latitude 24° 



07', adapted to a ratio of axes 299 to 300. 



