22 On Practical Geodesy. 



|^p° Referring to tlie approximate relation — 

 sin r _ sin L" 

 sin L' sin I" 



made use of in arriving at the preceding values of 8 , 8^^, it 

 may be proper to observe that we must not always use it as 

 if it were rigorously true. If so used we should, as a con- 

 sequence, have — 



sin A, _ sin D,^ 



sin D^ sin A.^ 



and therefore the first side of this equation always less than 

 unity, which we know to be absurd. Hence we perceive 

 that the adoption of the above approximate relation is 

 equivalent to assuming that between the limits of the 

 possible values of A^ from the state in which A^ = D^^ to 

 that in which A^ = V, we have sin D^ = sin A^, and sin 

 A^^ = sin D^^ so nearly true that their logarithms are the 

 same to 7 places of decimals. However, we will now shew 

 how those small angular differences can be computed. 



26. It is evident that the amount by which the angle A^^ 

 exceeds D^^ is truly expressed by the spherical excess of the 

 smaU triangle S^S^^D^^. It is also evident that the amount 

 by which the angle D^ exceeds A^ is expressed by the 

 spherical excess of the small triangle S^S^^D^. Hence (see 

 formula 4, page 158, Serrets', &c.) — 



cot J A,, = cot I D, 



tan 1 A,, = tan J D,, 

 tan i A, = tan J D, 

 cot J A, = cot ^ D, 



cos 1 (z, 



+ K) 



COS J {z, 

 COS 1 {z, 



-K) 

 -s») 



COS i {z, 

 COS J (z,^ 



+ S„) 



COS ^ {z,, 

 COS J (z^, 



-S,) 



(a.) 



COS 1 {z,, + 8,) 



We have also (see formula 3, page 158, of Serrets' Trigo- 

 nometry) rigorously — 



tan , (A, DJ _ i_tanl,^tan'l8.,cosD, (90) 



tan 1 (D. - A,) - ^^"^ * ''' *^^ * ^' '''' ^' 



1 + tan f z,^ tan J 8, cos D, 



And the angles J (A^^ — DJ, J (D — A^), being but 

 fractions of a second ; and the values of tan J z^ ' tan J S^^ 



