On Prctdical Geodesy. 7 



Hence also (since each of the four arcs is less than 90°) we 

 have 



sin I" 7 sin L" 



sin L" V sin L' (i s) 



sin L' 7 sin I' 



8. From the spherical triangles D.PS,,, D,,PS,, we have — 



sin L' sin D, = sin I" sin A,, 

 sin L" sin D,, = sin I' sin A, 



sin D, 7 sin A,, , x 



sin A, 7 sin D,, ^^^' 



And since each of the angles (D, + A,,), (A, + D^J, is less 

 than 180°, it foUows that— 



D^ 7 A^^, and that A^^ is acute / x 



A, 7 D,, and that D,^ is acute * '^''^ 



9. We shall now establish the following important rela- 

 tions between the azimuths and angles D,, D,, — 



D, 7 A, 



A, 7 A„ (21) 



A, 7 D, 



First, from the triangles S^PD,,, S,,PD,, we have — 



sin 2, sin A, = sin L" sin w 

 sin 0,^ sin A,, = sin L' sin 00 



But from (14), (15), and (16), it is evident that — 



z,, 7 z, (22) 



And therefore, since sin L" is greater than sin L' we have — 

 sin z, sin A, 7 sin z^^ sin A,, 

 sin A, _ , 



sin A,, 



Now, since A, + A,, is less than 180°, and that angle A,, is 

 acute (see 20), therefore it foUows that — 



A, 7 A^^ 



In order to shew that the first and third of the relations 

 (21) a,re true, we may proceed thus — 



Applying formula 4, page 158, of Serret's Trigonometry 

 to the spherical triangle S JS,,, and putting c to represent the 

 spherical excess of this triangle, we have — 



sin 1 (a, — a,,) 



tan i (a — c) = • tan J a (23) 



cos i (a, + a,,) 



