On Practical Geodesy. 



23 



cos D^^, and tan J z^^ ■ tan h 8, ' cos D^ being so extremely 

 small, it is evident we can Und the values of the angles A^^ 

 and A^ to the t o o o o P^^^ of a second by means of the amelio- 

 rated formulae — 



tan J (A^^ — D,J = sin D,^ tan ^ 0, * tan J 8,, , . 



tan 1 (D, — A j = sin D, tan J 2,, • tan J 8, ^^ '^ 



We can also arrive at these in the following manner — 



From formula (1), implicating spherical excess, on page 

 157 of Serrets' Trigonometry, we have — (since in actual 

 practice of surveying the logs, of cos J v, cos J z^, cos J z^^, 

 are the same to 6 or 7 places of decimals) — 



sin i (A,, — D,,) = sin D,, • tan J 2, • sin J 8,, , v 



sin i (D, — Aj = sin D, • tan J 2,, • sin 4 8, ^^'^ 



.*. also A^^ — D,^ = sin D,, tan J 0, * 8,, / x 



D, — A, = sin D, tan J 2,, • 8, ^''>' 



or, A, — D^^ = 1 . ,^ ■ 8^^ . sin 1" • sin D, 



D, — A, = J • 2,, • 8/- sin 1" • sin D, 

 And from these and formulae (87) and (88), we easily find — 



A,~D. 



= i 

 = i 



D. - A, = I 



= i 

 = i 



1 — e'^ 



sin I' ' sin L" sin (A, — D,,) • 0,^ X sin 1" 



l — e' 



e- 



. sin^ I 



^ ^ sin A, sin (A, — D 



sin D, 



^•2/x sinl" 



. ,, sin A, sin (A, — D,,) 



sin V ' '- — -^^ — '^ '-^ ' 



sm (o 



z? X sinn" 



1 — e' 



' sin I" • sin L' • sin (D, — A J • z,,^ x sinl " 



1 — 



v,2 



Sin 



,^„. sinA„sm(D,-A„) . ■^^, 



sin D, ' 



sin I 



„ ^ sin A,, sin (D, — A,,) 



2 ' X sin^ 1" 



1 — e" sm (0 



In the "Account of the Principal Triangulation of Great 

 Britain and Ireland " (see pages 248, 249, formulae 32 and 

 36), the following erroneous expressions are given — 



A, = i • ^-^, • cos^ I, sin 2A, • z\ x sm V 



(-) 



D. 



D, — A, = J 



1 —e' 



o2 



e" 



/ cos^ l,^ sin 2A,, * s^, X sin 1" 



with respect to which we may observe — 



1°. From them we should infer that D^^ — A^^ and D^ — A^ 

 have finite values when the latitudes of the stations are 



