98 RELATIONS PERTAINING SIMPLY [BOOK I. 



If the time, to which the computed place corresponds, is supposed to be 

 distant n days from the epoch, and the mean longitude for the epoch is 

 denoted by N, the daily motion by T, we shall have M N -\- nt -- IT, and thus 

 d M = d N-\- ndf dI7. In our example, the time answering to the computed 

 place is October 17.41507 days, of the year 1804, at the meridian of Paris: if, 

 accordingly, the beginning of the year 1805 is taken for the epoch, then 

 11= - 74.58493; the mean longitude for that epoch was 4152 21 / .61, and the 

 diurnal motion, 824&quot;.79S8. Substituting now in the place of d M its value in 

 the formulas just found, the differential changes of the geocentric place, expressed 

 by means of the changes of the elements alone, are as follows : 



&l = 2.41287 A.N 179.96 dr 0.75214 d/7 3.00531 dy -f 0.16488 da 



- 0.11152 dt-f 0.04385 d8, 



AI-- - 0.66572 &N+ 49.65 dr -f 0.23677 d J7 + 0.61331 dq&amp;gt; -f 0.02935 da 



- 0.47335 di-f 0.38090 da. 



If the mass of the heavenly body is either neglected, or is regarded as 

 known, r and a will be dependent upon each other, and so either dT or da may 

 be eliminated from our formulas. Thus, since by article 6 we have 



we have also 



dr _ 3 da 

 T * a 



in which formula, if dr is to be expressed in parts of the radius, it will be neces 

 sary to express r in the same manner. Thus in our example we have 



log* . . . . . 2.91635 



logl&quot; 4.68557 



logf 0.17609 



C.loga .... 9.57756 



7.35557, 



or, dr = 0.0022676 da, and da = -- 440.99 dT, which value being substituted 

 in our formulas, the final form at length becomes : 



