SECT. 2.] TO POSITION IN SPACE. Go 



x = ar sin (M + 248 8 55 22 / .97) = ar sm\u -j- 2G3 47 35&quot;.39) 

 y = 5rsin(2 -fl58 5 54 .97) = br sin (M + 172 58 7.39) 

 z = crsinu =eram(u-\- 14 5212.42) 



in which log e = log sin { = 9.3081870. 



Another solution of the problem here treated is found in Von Zach s Monatliche 

 Corresponded, B. IX. p. 385. 



57. 



Accordingly, the distance of a heavenly body from any plane passing through 

 the sun can be reduced to the form krsm(v -\- K}, v denoting the true anomaly; 

 k will be the sine of the inclination of the orbit to this plane, K the distance 

 of the perihelion from the ascending node of the orbit in the same plane. So far 

 as the position of the plane of the orbit, and of the line of apsides in it, and also 

 the position of the plane to which the distances are referred, can be regarded as 

 constant, k and K will also be constant. In such a case, however, that method 

 will be more frequently called into use in which the third assumption, at least, is 

 not allowed, even if the perturbations should be neglected, which always affect 

 the first and second to a certain extent. This happens as often as the distances 

 are referred to the equator, or to a plane cutting the equator at a right angle 

 in given right ascension: for since the position of the equator is variable, owing to 

 the precession of the equinoxes and moreover to the nutation (if the true and not 

 the mean position should be in question), in this case also k and K will be subject 

 to changes, though undoubtedly slow. The computation of these changes can be 

 made by means of differential formulas obtained without difficulty : but here 

 it may be, for the sake of brevity, sufficient to add the differential variations 

 of/, Q, and //, so far as they depend upon the changes of & n and e. 



d* w = sine sin8 d(8 n) cosS de 



sin i cos A , , x , sin Q , 



C 



sin i sin r 



Finally, when the problem only is, that several places of a celestial body with 



9 



