SECT. 2.] TO POSITION IN SPACE. 81 



earth, can be determined directly with the greatest facility ; but as in the com 

 mon method, given in article 62 and the following articles, the geocentric place is 

 commonly referred to the centre of the earth, in which case it is said to be free 

 from parallax, it will be necessary to add a particular method for determining the 

 parallax, which is the difference between the two places. 



Let the geocentric longitude and latitude of the heavenly body with reference 

 to the centre of the earth be X, ft ; the same with respect to any point whatever 

 on the surface of the earth be I, b ; the distance of the body from the centre of 

 the earth, r; from the point on the surface, z/; lastly, let the longitude L, and the 

 latitude B, correspond to the zenith of this point in the celestial sphere, and let 

 the radius of the earth be denoted by R. Now it is evident that all the equations 

 of article 62 will be applicable to this place also, but they can be materially 

 abridged, since in this place R expresses a quantity which nearly vanishes in 

 comparison with r and A. The same equations evidently will hold good if K,l,L 

 denote right ascensions instead of longitudes, and ft, b, B, declinations instead of 

 latitudes. In this case / X, b ft, will be the parallaxes in right ascension and 

 declination, but in the other, parallaxes in longitude and latitude. If, accord 

 ingly, R is regarded as a quantity of the first order, I I, b ft, /I r, will be 

 quantities of the same order ; and the higher orders being neglected, from the 

 formulas of article 62 will be readily derived : 



L, . __ Rco^BsmQ. L) 

 I A - 



r cosp 



n . g-8= 



HI. J r= R cos B sin ft (cotan ft cos (X L] -\- tan Bj. 

 The auxiliary angle 6 being so taken that 



tan B 



tan B = 



- - j^, 

 cos (i L) 



the equations II. and III. assume the following form : 



-p, , _ B cos B cos (). Z) sin (|3 0) __ .K sin 2? sin (5 0) 



r cos r sin 



-pp. . R cos B cos (I L) cos (|3 6) _ R sin B cos (0 0) 



Mi, /t - T - - - - -- ; - . 



cos sin a 



11 



