6 On Practical Geodesy. 



. sin «/ _ I^// /^ \ 



sin a,^ R^ 



And since we suppose I, greater than l,„ we know that R, 

 is greater than R,, ; and hence we learn that the angle of 

 depression a,^ adjacent to the station having the lesser 

 latitude is greater than the angle of depression a, adjacent 

 to the station having the greater latitude. 

 6. We have, evidently — 



or, which is the same — 



tan a, tan a,, 



tan (z, — a,) tan {z,^ — a,) 



(.a) 



Now it is evident that each side of this equation is greater 

 than unity; and .'. when z, and z,, are each less than a 

 quadrant, we have — 



a, 1 z, — a, / X 



«// 7 Z,, — a,, ^ ^ 



7. If the latitudes l„ l,„ of any two stations (on the same 

 side of the earth's equator) be of constant magnitudes, then, 

 no matter how otherwise the stations may vary in position, 

 it is evident that the points Z^, Z^^, in which the normals 

 cut the polar axis, remain fixed. It is also evident that as 

 regards the magnitudes of L', L", 8^, 8,,, they too are con- 

 stants, and the same as if the stations were on one meridian. 

 Hence it is obvious that when l^ is greater than l^^, or, which 

 is the same — when I" is greater than l\ we know that the 

 first and third of the following are true — 



I" - L" 



L" 7 L' (17) 



L' 7 I' 



The truth of the second of these relations is easily seen. 

 For drawing perpendiculars S^H^, S^^H^^, from the stations 

 to the polar axis, it is evident we have — 



tanL" = S,,H,, ^ (Z,,H,, + Z,,Z,) 

 tanL' = S,H, - (Z,,H,, + H,,HJ; 



and therefore since S^^H^^ 7 S„H^, and that Z„„Z„ l H^^H^, 



tan L" 7 tan L' 

 L" 7 L' 



