102 



ON THE FORMULAE FOR CALCULATING 



Likewise in the latter case, which takes place when before the maxi- 

 mum in the Eastern and after the maximum in the Western Elongation, 

 if we put 



/tan SS\ tanX. sin PSS'\^ . ^, 



I ~ — tan 6' 



\ cos Z J 



cos" PS. tan^h P . „ 



Then tan h Z — 



sin X 



* 1 ^ COS- PS. tati i § P. sin- 6 



And h Z z=: : -. — -j, — 



sm X. sm V 



This method is quite rigorous, but it is rather more operose than the 

 nature of the case usually requires : before, however, proceeding to sim- 

 plify the formulae, it may not be worth while to give an example of each 

 of the cases above adverted to, and to that end let it be required to deter- 

 mine what would be the correction to be applied to the Azimuth, if instead 

 of the actual instant of maximum on the 4th May 1830, the star had 

 been observed thirty minutes before or after that occurrence. 



This computation will be most conveniently arranged according to 

 the following form, premising that, in deducing the tangent of a very 

 small arc from its sine, and vice versa, the easiest method is to add or 

 subtract the Log. Secant, and that to deduce the tangent of an angle 

 from the sine of half the angle, the easiest way is to add to the latter the 

 Log. of 2, -|- 3 times Log. Secant; as is evident from the following con- 

 sideration : _ 



tan 'I 6 — — 2 tan 6. (1 -f tan" d -j- tan &\ &c.) 



:. tan2 — 2 sin 6. sec ^. (1 tan" = 2 sin 0. sec^ 6 



