by observations of their Depression. 183 



It is obvious also that 



a" = (0-5 - n) — K = (0-5 - n)/K, . . . (5.) 



in which jT is the factor to convert feet on the earth's surface 

 into seconds of arc in any given direction. 



2. Formula (1.) fails to determine A" with sufficient accu- 

 racy when the height h is considerable, and the distance K 

 extends to, or near to, the visible horizon. 



In this case we may proceed as follows : — 



B C : A C : : sin H A C : sin A H C = — sin H A C. 



Whence 180° -(AHC + HAC) = HCA, 

 and £HCB = HFB = IBH = A". 



Wherefore AIB-IBH = AHB; 



g : g + h : : cos D : sin A H C = (l + — ) cos D . (6.) 

 But h being very small in comparison with g, 1 -\ exceeds 



unity by a very small fraction, therefore log ( 1 H ) = — , 



in which /* is the logarithmic modulus. If g be taken equal to 



the radius of the equator as a mean value, then log ( 1 -{ ) 



— 8*3171G78. Instead of the mean value of g being taken as 

 above, 



logo = log R" — log /exactly . . . . (7.) 



Hence log sine A H C = log cos D + — h . . . (8.) 



always obtuse, of which in this problem its excess above 90° 

 is taken. 



In the preceding formula it is supposed that the coefficient 

 of refraction, w, is determined by observation or calculation. 

 The mean value, however, amounting to 0'08 of the inter- 

 cepted arc, will frequently be sufficient. In this case the effect 

 will be 0*08 K, or 0*16 A", a quantity to be added to D while 

 A" must be subtracted. 



Hence D - A" + 0-16 A" = D - 0*84 A" = D - a", 

 by making a" = 0*84 A". 



Recurring to formula (1.), by substitution we have 



log a" = const, log 7'6 170579 + log cotD + log^ . (9.) 



But a" = —?— a" = !£?? a" = 0- 1 904-762 a" = \- a" nearly. 

 0'5 — n 0-42 5 ' 



