On Practical Geodesy. 15 



But (14) sin a, R,, . . 



tan a, _ cos I, sin A, . ^ 



tan a,, cos Z,, sin A,, 

 From these we can easily express the squares of the sines, 

 cosines, and tangents of the angles of depression of the 

 chord in terms of the two latitudes and two azimuths ; but 

 it is obvious that such expressions must assume the inde- 

 finite form % when the latitudes are equal, or R, = R,^. 

 And from (64) and (27), we have — 



, i / I \_ /^/ + ^//V l^n ^^^ ^// ^^ -^'1 — ^/ ^^^ K sill AAi 

 tan 2 ^a, + a,)-\^^-— ^y • \^^ ^^^ ^ ^ ^.^ ^ _^ ^ ^^^ ^^ ^.^ ^f 



tan 1 (a —a\-( ^'~^" Y' Z -^// cos I,, sin A,,— R, cos I, sin A, U 

 2 V - 'J \_R^ _|_ ^J Ve,^^ cos Z,, sin a,, + R, cos I, sin A/ 



The expression for tan J iS or tan \ {a^^ + o&J, given in (67), 

 is of a hke character. It assumes the indefinite form % when 

 R, = R,^; which is the case on a spheroid when the latitudes 

 of the stations are equal, and always the case on a sphere, no 

 matter how the stations may be situated with respect to 

 each other. 



23. From the triangles D^SJ, B^^S.J, we have — 

 cos a, sin D, 



cos (z„ — a,) sin A, 



cos a„ __ sin D,^ 

 cos (z, — a,) sin A,^ 



. -r^ COS I,, sin (0 



sm D, = :^ 



sin D„ = 



sin z, 

 cos I, sin <o 



(eO 



(-) 



(71) 



And from these we at once obtain the relations — 



. sin A,, cos a,, 



cot z, = — r^ "— __ tan a, 



cos l, sm 0) cos a, ' 



. sin A, cos a, , 



cot z.. = — r^ — — tan a,, 



cos l^ sm <o cos a,, 



If in these we substitute the values of sin w from (60) we 



have — 



. ^ • cos a, 



tan z, = , . ' 



R, — A; sm a, y v 



(72) 



. A; • cos a,, 



tan z,, = Ji 



XV,, — /c ' sm a,. 



