58 On Practical Geodesy, 



And to find the arcs e^^, f^^, whose sum = 8,^ ; we have^ — 



tan i (e,. -/„) = Jj^i^^ ; i (e., + /„) = J 8„ 



From these we easily obtain the values — 



e,^ = 7-75773 /, = 7-75729 



e, = 7-65453 / = 7-65497 



In the spherical triangle F^PF^,, we know the values of 

 the sides and included angle w ; and applying the usual for- 

 mulse we find — 



angle F, = 134°,, 44',, 02" • 79079 

 angle F„ = 44°,, 30',, 17" • 61004 

 arc F,F„ = 1°,, 24',, 19" • 02484 = J (2, + 2,,) 



.-. F, = J (A, + D,) to within 0"-0001 

 /. F,, = ^ (A„ + D,,) to within 0"-0002 



We may also observe that — 



D, — A, = 0"-13345 ; A„ — D„ = 0"-13336 



.-. D, — A, = A„ — D., to within 0"-0001 



In the "Account of the Principal Triangulation of 

 Great Britain and Ireland," the following formulae are 

 given — 



D, — A, = J • 5-7^ • cos^ ^„ sin 2 A„ • 0,,^ • sin 1' 



D„ — A„ = J • :j 2 ' cos^ I, sin 2 A, • «/ • sin 1" 



In working out these expressions with respect to the 

 present examples we have — 



log J = r-3979400087 log J = 1-3979400087 



log T-^ = 3-8345159915 log j-^—, = 3'-8345159915 



1 — 6 J. — 6 



cos^ Z„ = 1-8046972330 eos^ l^ = 1-7930642882 



sin 2 A„ = 1-9999812911 sin 2 A, = 1-9997379520 



log s„2 = 7-4081585260 log z,^ = 7-4081087226 



sin 1" = 6-6855748668 sin 1" = 6-6855748668 



.•.log(D,-A,) = 1-1308679171 .*. log(A„-D„) = 1-1189418298 

 .-. D, — A, = 0"-1352 which is too great by 0"-002 

 A,, — D„ = 0"-1315 which is too small by 0"-002 



We may also observe that in all cases in which the 

 greater azimuth A^ is less than 90°, the second of the above 



