152 GEODESIC IXTESTIGATIOKS. 



©If with I as pole we describe a great circle of tlie unit sphere S' to 

 cut the arcs S^D,^, S^^D,, tS^S^,, in U,, E^^, 0, respectively, it is evident 

 that — 



arc S, E, = i2' 

 arc S^,E,, = ^2" 

 Moreover, it is evident that for any pair of mutually visible stations S', S" 

 on the earth, we have — 



2 = i 2' .+ i 2" 

 And we may also assume — 



cos ^ 2' ^ cos i 2" ^ cos i 2, 

 without appreciable error in calculated results. 



This can be easily inferred from the following elucidation : — 



We have— S^E,= i^,' ; S,I = i tt — ^2'; 

 S^^E^, = i 2"; S,J = I TT 4- i2"; 



i2' + i2" = 2,• 

 i 2" 7 i 2'. 

 Applying formula (4) given by Serret, on page 158 of his " Traite de Tri- 

 gonometric," to the spherical triangle S,IS^„ and pvitting e for the spherical 

 excess of this triangle, we have — 



tan HA - e) == tan U- '''' lf,l~\'^J, 

 cos i (i 2' + i 2") 



And since we know that the numerator of the fractional portion on the 

 right-hand side of this equation is necessarily negative, .". we learn that the 

 angle A or S^IS^, is less than e, the spherical excess of the triangle S^IS^^. 

 Hence we infer that — 



angle IS^S^, -f angle IS^^S^ ~7 tt 

 and .". also — 



angle S^S^^D, V angle S,,S^D,, 

 and that — 



A' 4- A" -J angle P,S,S„ + angle FS,,S„ or "7 a' + a". 



And we perceive that the sum of the azimuths A', A" is greater than the 

 sum of the two angles a' , a", of the triangle S,PS,j by the amount which the 

 angle S^S^^D^ exceeds S^,S^D,, ; or, which is the same, by the amount which 

 e, the spherical excess of the triangle S,IS^„ exceeds A, or angle S^IS^^. 



The equation — 



f X ^ AN f X A gin i a 2" — i 20 



tan k (e — A) = tan * A. , ,, _„ — ; — , ' 



- ^ ' cos i (i 2" H- i 2') 



shows what a very small angle € — A must be. In actual practice, in which 

 the stations are not over 100 miles asunder, e — A will always be less than 

 TcuTo piii't of a second. 



cos k ^ _ sin S,S„T> _ ^ 



cos 2 2" sin S,^SjD 

 /<7N The following relations are also worthy of particular notice :— 



D, V A' 



A' V A" 



^" -7 D,, 



Their demonstration may be as follows : — 



The triangle S,ID^ is evidently such that, — 



angle IS,I), + angle ID,S, Z. tt 

 but angle PD,S„ + angle ID,S, — ir 



angle PD,S„ -7 angle IS,B 

 or D, -7 A'. 



