22 MEMOIRS NATIONAL ACADEMY OF SCIENCES. [Voi.xiv. 



Ii:sm(x+K)=- 70.'5 



Ji^cos (x+K)=+lQlf6 



<7, = + SS.'Q 

 (7j=+120.»8 



From Table LIV, multiplied by i cos i we have three terms in 



Ik sin (x+ K)=- 4.''2 - l.^O sin e + 2f7 cos s 

 which, added to 



(7, (cos £-€,) + (72 sin £ =+ 1 20.''8 sin j + 35:'9 (cos £ - ej 



gives for two terms in lie sin (x+ K) 



-T.'S + llS.'gsin £ + 38^6 cos £=(0.89) sin 270?0+ (2.0970) sin (£ + 17?99) 



CHECK COMPUTATION. 



After the elements have been determined and the final tabulation of the perturbations is 

 ready, the following checks should be performed, even if the computation has been duplicated. 



t = 



(?o = i(£i-«'isin£,)-^' 



1 — w 

 g' = c' + [n'dz'] Oo = d„+ —^ {ndz - [ndz] ) - ijw sm £ 



where the necessary quantities are to be taken from the last approximation for c. 

 Secondly, the heliocentric coordinates 



x — Jx, y — Ay, z — Jz 



for < = must check when computed by the usual formulae for two body motion and osculating 

 elements, and when computed with the final set of elements and the corresponding perturba- 

 tions, 71^2 and V, taken from the final tabulation. 



The final tabulation of the perturbation in the third coordinate is checked by the test 



t = ; M = 



COMPUTATION OF THE PERTURBATIONS FOR THE TIME t. 



It is well to emphasize here the distinction between the elements n^ and c, and the elements 

 7^2 and Cj in their relation to the perturbations. Let 7)52, denote the perturbation in the mean 

 anomaly computed according to equations (20), (19), (IS), (21), (22), (27), (28), and let 71^22 

 signify the pertiu-bation computed according to equations (20), (19), (18), (22), the final tabu- 

 lation of {ndz — [ndz]), and an equation analogous to (34). (It must be remembered that equa- 

 tion (34) is for {10) Hygiea only. The numerical coefficients are determined for each ]>lanet 

 individually.) 



Before the determination of c there can be no confusion, for there is but one way to com- 

 pute the perturbation ndz. Later, when both c, and c^ are given, the computation may be per- 

 formed in either manner. The latter method is, of course, adoj>ted. The question then arises, 

 what values of £ and c are to be used in equation (19) ? 



