66 RELATIONS PERTAINING SIMPLY [BOOK I. 



respect to such variable planes may be computed, which places embrace a mod 

 erate interval of time (oue year, for example), it will generally be most con 

 venient to calculate the quantities a, A, b, B, c, C, for the two epochs between 

 which they fall, and to derive from them by simple interpolation the changes for 

 the particular times proposed. 



58. 



Our formulas for distances from given planes involve v and r ; when it is 

 necessary to determine these quantities first from the time, it will be possible to 

 abridge part of the operations still more, and thus greatly to lighten the labor. 

 These distances can be immediately derived, by means of a very simple formula, 

 from the eccentric anomaly in the ellipse, or from the auxiliary quantity F or u 

 in the hyperbola, so that there will be no need of the computation of the true 

 anomaly and radius vector. The expression kr sin (v -\- K] is changed ; 



I. For the ellipse, the symbols in article 8 being retained, into 



ak cosy cos JT sin E-\- ak sin K (cos E e). 

 Determining, therefore, /, L, X, by means of the equations 



aksin K= IsinL 

 ak cos (f cos K=l cos L 



.K=i el 



our expression passes into I sin (E -f- L) -\- X, in which I, L, &quot;k will be constant, so 

 far as it is admissible to regard k, K, e as constant ; but if not, the same precepts 

 which we laid down in the preceding article will be sufficient for computing their 

 changes. 



We add, for the sake of an example, the transformation of the expression for 

 # found in article 56, in which we put the longitude of the perihelion = 121 17 

 34% 9 == 14 13 3r.97, log a = 0.4423790. The distance of the perihelion from 

 the ascending node in the ecliptic, therefore, = 308 49 20&quot;.7 = ti v; hence 

 K= 212 36 56&quot;.09. Thus we have, 



