SECT. 2.] TO POSITION IN SPACE. 67 



log a/c ..... 0.4411713 log I sin L .... 0.1727GOOn 

 logging .... 9.7315887 n log I cos L . . . . 0.3531154 n 

 log a k cos (f . . . 0.4276456 whence L = 21325 51&quot;.30 



log cos K .... 9.9254698 n log^ = 0.4316627 



logJl= 9.5632352 



1= +0.3657929. 



II. In the hyperbola the formula k r sin (v -\- K), by article 21, passes into 

 X -}- ju tan F -\- v sec F, if we put e b It sin ZT I, b k tan if cos K= /A, bk sin K 

 = v ; it is also, evidently, allowable to bring the same expression under the form 



nsm(F-\-N)-{-v 

 cosF 



If the auxiliary quantity u is used in the place of F, the expression /crsin (v-\-K] 

 will pass, by article 21, into 



in which a, ft, y, are determined by means of the formulas 

 a = 7, = e b k sin K 



y = (v jtt) = ebk sin 



III. In the parabola, where the true anomaly is derived directly from the time, 

 nothing would remain but to substitute for the radius vector its value. Thus, 

 denoting the perihelion distance by q, the expression kr sin (v -f- -ff&quot;) becomes 



q k sin (v -\- K) 



59. 



The precepts for determining distances from planes passing through the sun 

 may, it is evident, be applied to distances from the earth ; here, indeed, only the 

 most simple cases usually occur. Let R be the distance of the earth from the sun, 

 L the heliocentric longitude of the earth (which differs 180 from the geocentric 

 longitude of the sun), lastly,^, Y, Z, the distances of the earth from three planes 

 cutting each other in the sun at right angles. Now if 



