18 On Practical Geodesy. 



Moreover, it is evident that by putting Y to repre- 

 sent the particular value of the angle A, when unequal to D^ 

 but such that sin A, — sin D, (m which case A, is acute and 

 D, obtuse) it is evident that — 



whenever A^ 7 Y, then will z^^ l a„ -\- a^ ot% 

 whenever A, l Y, then will z,, i a,, -\- a, or 2 



Hence : — If S^^ be any fixed point within any convex closed 

 curve on the earth's spheroidal surface, and Z^o the point in 

 which the normal to the surface at Soo cuts the polar axis : 

 then there are 4 real points S^ on this curve, and 4 only, 

 such that the angle S„„Z^^S„ subtended at Z^„ is equal to 

 the sum of the angles a^^, a^, of depression of the chord S^^^o 

 below the tangent planes at S^^? ^o- ^iz. — The two points 

 in which the curve is cut by the plane X through S„„ which 

 is perpendicular to the polar axis ; and the two points lying 

 on the same side of X, and such that the azimuth of S^ taken 

 at S^Q is acute, and the azimuth of S^^ taken at S^ is also 

 acute but gTeater than the other, and approacliing very 

 nearly to 90° owing to the earth's small ellipticity. 



24. From the triangles S^^PB^, S^PD,,, we have- 

 sin z„ sin A„ 



sin L' 



sin 0) 



. T // sin z, sin A, 



sm L" = '- ' 



sin o) 



cos L' = cos z,, cos I" 4- sin z,, sin I" cos A,, 

 cos L" = cos z, cos V 4- sin z, sin V cos A, 



, T- / cot A,, sin (0 + cos I" cos w 



cot L = r—i 



sm I" 



, T „ cot A, sin w + cos V cos w 



cot L" = '- ! 



sm I' 



(,s) 



And since L' and L" are the circular measures of the angles 

 between the lines S^Z^^, S^^Z^, and the polar axis, we have 

 evidently — 



cot L' = e'- ^" ^^^ \' + (1 — e') tan I, 



K, cos t, / V 



(79} 



cot L" = e^ ' =-^ -' + (1 — e^) tan I.. 



R,, cos l,j 



25. By letting fall perpendiculars from Z^, Z^, on the 



