and Declifiation into Longitude and Latitude. 331 



sin 8 = sin e cos (3 sin A + sin |3 cos e 

 cos 8 sin a = cos s cos /3 sin X — sin ^ sin e ('i) 



cos 8 cos ex. = cos jS cos X 



They are general through the whole circumference of the cir- 

 cle, inasmuch as, by the nature of the case, cos « and cos A 

 have always the same signs, or as a. and A are always at the 

 same time between 90° and 270°, or 270° and 90°. 



For facilitating the logarithmic calculation it is usual to in- 

 troduce an auxiliary angle, by means of which the two parts 

 on one side ax-e changed into one only. For (1) put: 

 sin 8 = M sin N 

 cos 8 sin a = M cos N 



hence sin |3 = M sin (N — e) 



cos /3 sin A = M cos (N — e) 

 cos /3 cos A = cos 8 cos u 



The value of M from the two first auxiliary formulas being 

 substituted in the latter three ; viz. M = -^^1--^—^ we obtain 



cos N 



tang N = ^^ 



" sin as 



COS (N — s) 



tang (3 = tang (N — e) sin A 

 to which may be added as a check on the calculation 



cos(N— s) cos /3 sin X 



cos N cos S sin a 



These formulfE give every thing by means of tangents, con- 

 sequently in the most accurate manner; and (with due regard 

 to the remark above made) without any ambiguity ; the geo- 

 metrical signification of M and N will be easily found. For a 

 single calculation they are beyond doubt the most convenient; 

 but they do not admit of a table of single entry even for con- 

 stant e, and have besides the small disadvantage that for con- 

 verting at the same time several places with regular differences, 

 the auxiliary angle N is, in the vicinity of 0"" and 180°, more 

 irregular than tijc given principal quantities. 



For the system (2) we have, in like manner : 



tang N' = ^«A 



" sin X 



cos N' + t . 



tanji: « = --, — tang A 



" cos N " 



tang 8 = tang (N'-f-:) sin a 



cos N' -f I cos i sill a. 



cos N COS /5 bin A 



2 I ' 2 The 



