132 Solutions to Important Problems. 



b have been measured or otherwise obtained : for the angle /? 

 (which is also constant for all positions of the point of observa- 

 tion) can be easily found with accuracy from its "tangent, &c. 



Case 2. 



If the segments a and b be adjacent, and x one of the 

 extremes (the segments being in the order c, b, x, ) ; then A, B, 

 C being the angles which a, b, and x subtend at P, we have 



I (a + b+x) — sinB . sin (A+B + C) 

 a x sin A . sin C 



a constant determinable number for all positions of the point of 

 observation P, which we may denote by n. 



Hence x = — - 1 



n a — b 



If we find 8 such that 



Sec 2 B = a • s i n A . sin (A + B + C?) r 2 (5) 



b . sin A . sin C 



then x = (a + b) r2 (6) 



tan2 B v ' 



And the ameliorated formulas when a, b, x are circular measures 

 of portions of a geodesic, are obviously 



o , n sin a . sin A. . sin (A + B + C) r% ,» x 



Sec 2 B = . . . v . - '- — (7) 



sin b . sm A . sin (J 



. Ci±_ 6 ) . cos (£=?) 

 = 2 sin V 2 / V 2 / 



tan 2 /3 



(8) 



Vebification of measueed portions of a Base Line. 



If", b, c be three consecutive portions of the trace of any 

 geodesic of the earth on the reducd spheroidal surface, and that 

 A, B, C be the dihedral angles formed by the pencil of normal 

 planes through any point P, and the extremities of a, b, and 

 c (we suppose the normals through the extremities of a, b, and c 

 to meet the normal through P in the same points), we have shewn 

 how lo find any of the three portions in terms of the other two 

 and the angles A, B, C. 



