30 On Practical Geodesy. 



and .'. 



^ . cos I, COS A,+cos l,^ cos A,, ^ , — , — - — -. — — - 



tan h^ = - — H^ — ^ — A — ; — ■ — i^ — '■ — a~' " Jcos I, cos I,, sin A, sm A,. 

 ^ sm 2^,sm A, + sm2 ^,, sm A,, v / // / -y 



The expressions given for tlie tangents of the angles of 

 depression of the geodesic chord in (110) and (111) implicate 

 the assumed eccentricity of the earth, while the expressions 

 (121) depend entirely on the observed latitudes and azi- 

 muths. If applied to the example 1 problem 1 given in the 

 sequel (which may be regarded as an extreme case in trigo- 

 nometrical surveying) it will be found that the resulting 

 values of a, and a,, can be accurately determined to -^^^q-q 

 part of one second, — their logs, holding true to 8 places of 



decimals. 



■p -p , 



By substituting in (111) the values ^ and ^ as given in 



(51), we easily rearrive at formulae (121) ; and by like 

 substitutions in (110), we easily find the following values 

 for the tangents of the angles of depression of the chord — 

 true to at least 8 places of decimals in their logs — 



. sin A,, sin A, cot w -\- cos A, sin I, 



tan a, — " ' ' ' ' 



tan a,, — 



cos l^^ sin 0) " 



And when a,^ and a, are found, we have % = a,, + a,. 



However, there are other methods of finding approximate 

 values of %, in terms of the latitudes, azimuths, and length 

 of arc between the stations, &c. ; but I defer their con- 

 sideration for a future paper. 



37. With respect to the figure it may be observed that if 

 F^ and F^^ be the points in which the chordal plane NS^S^^ 

 cuts the arcs PS^, PS^^, it is evident that the arc PF^ is 

 divided harmonically in S^, D^, and that the arc PF^^ is 

 divided harmonically in D^^, S^^. For the anharmonic ratio 

 of the points PF^S^D^ is the same as that of the pencil of 

 straight fines S^ • (PF^S^DJ, and /. the same as that of the 

 four points oo, N, Z^, Z^^, in which oo represents the point at 

 infinity in which the line S^P cuts the line CZ^Z^^, &c. 

 Hence the spherical pencil I • (PF^S DJ is harmonic. 



cos I, sin (0 cos I, 





sin A,, 





cos C^ sm (0 ' 





sin A^ sin A,^ cofc w + cos A,, 



sin^,, 



cos Z,, sin 0) cos l^, 





sin A, 





