Solutions to Important Problems. 133 



As security against errors of measurement in the two arcs, and 

 as a check on the calculations, we have the following anharmonic 

 equations of verification, whose right hand members can — by 

 using a first class altitude and azimuth instrument — be obtained 

 with accuracy to any desired number of 6gures, as they are func- 

 tions having constant magnitudes for all positions of the point of 

 observation P. 



sin {a + b) . sin (b + c) sin (A + B) . sin (B + C) 



sin b . sin (a + b + c) sin B . sin ( A + B + C) 



s in a + b ) . sin (b + c) sin (A + B) . sin (B + C) 



sin a . sin c sin A . sin C 



It is evident from these equations that if a, b, c be three conse- 

 cutive measured arcs of a long Base-Line, on the correctness of 

 which implicit reliance could be placed, then we should be able 

 to find out the closest approximation amongst assumed lengths 

 for the radius of curvature in the vicinity of the base-line : for 

 the measured dihedral angles A, B, C at any station remain the 

 same, no matter what length maybe assumed as the radius ; while 

 the functions of the arcs on the left hand sides of the equations 

 vary with the lengths we assume for the radius ; and therefore 

 the equations should approach the more nearly to absolute veri- 

 fication according to the approximate correctness of the assumed 

 lengths. However, it may be observed that small differences in 

 the assumed radii generate extremely small differences in the 

 corresponding values of the functions of the arcs which are the 

 left hand members of the above equations. 



Conyerselt. — If (as is always assumed), we have a near ap- 

 proximate to the length of the earth's radius of curvature in the 

 vicinity of a measured base-line, we can by the method above 

 indicated — which is remarkably expeditious, and independent of 

 lineal measurements and allowances for spherical excess — test the 

 existence of uniform accuracy in the measurements of its various 

 portions. For if a, b, c be three consecutive measured arcs of 

 the base-line, and that they do not fulfil the conditions imposed 

 by the above equations whose right hand members can be deter- 

 mined with precision to any number of figures (being constants 

 for all positions of P), then there must be an error in the mea- 

 surement of such base-line. 



