178 GEODESIC INYESTIGATIONS. 



We have also the following set of formulse rigorously true for any sphe- 

 roidal triangle on any spheroid or any analogous triangle on any surface 

 whatever- 

 cos K-^ = sin i;8" sin \y' -f- cos |/3" cos \y' cos A '\ 



cos K2 = sin iy" sin ia' -j- cos iy" cos ia' cos -S > (5) 



cos Kg = sin ia" sin JjS' -f cos ia" cos i/8' cos C ) 



The values of the chordal angles obtained from both sets of equations should 

 be equals if the work has been correctly performed ; and we should also have 



Xj -j, X2 -f ^3 = 180°. 



If these equations or relations subsist to within an extremely close degree of 

 absolute accuracy, they are (in conjunction with the tests already afforded) 

 sufficient to enable any competent mathematician to state, with confidence, 

 that the magnitudes of the geodesic arcs, chords, and angles have been found 

 with the greatest accuracy, and that the relative astronomical positions and 

 mutual bearings of the stations have been determined with precision. 



It is evident that the differences of the spheroidal and chordal angles of 

 any spheroidal triangle on any spheroid whatever, are rigorously expressed by 



.riaiigie uu auy spueroiu wnaLcvei, are zugorousiy expresseu uy 

 cos-^ (sin i/3" sin ^y' -\- cos i;8" cos \y' cos A)~\ 

 ^ — cos-i (sin \y" sin \a' -\- cos \y" cos \a' cos jS) \ ... (e) 

 C — cos-^ (sin \a" sin \^' -j- cos \a" cos \^' cos C)j 

 The sum of the three expressing the spheroidal excess. 



%^ I may here observe that if the process just indicated, with respect to 

 three chosen stations, be applied to any three mntually visible stations 

 forming the vertices of a great primary triangle of a Trigonometrical Survey, 

 it will be found to be an efficient method of exposing the errors due to the use 

 of erroneous formulae and the misrepresentations of the bearings and positions 

 of stations which may possibly result from permitting the officer who makes 

 the principal astronomical observations to have any control over the necessary 

 amount or the character of the computations ; for computations, when rigor- 

 ously and efficiently performed, are certain to add so considerably to the 

 amount of work required from the chief astronomical observer as to keep him 

 constantly in the field supplying the necessary data, and testing his own and 

 subordinates' work, whenever such is reported to be too incompatible by the 

 computers. 



The method of testing the accuracy of a survey by means of a " base of 

 verification" of a few miles in length is thoroughly deceptive when the primi- 

 tive base and base of verification are nearly in the same latitude ; for a 

 compensation of errors is very likely, in such case, to make the computed and 

 measured lengths of short lines to closely agree, even though the usual 

 erroneous and inefficient formulae be used in the computation. If the bases 

 differ extensively in latitude and longitude, it is probable that, under like 

 circumstances, the measured and computed lengths will differ by some small 

 amount. But, in this case also, the test is thoroughly inadequate as regards 

 the lengths and bearings of the sides of the great primary triangles, and par- 

 ticularly so as regards the astronomical or geodesic positions of their vertices, 

 which is a matter of much more consequence, when looked at from either a 

 practical or scientific point of view. 



It is scarce.y necessary to add tnat, by selecting a set of connected triangles 

 extending tln-ough the triangulation of any extensive country, we could, by 

 such means, test the general acciiracy of the entire work. 



