No. 3.1 MINOll PLANETS— LEUSCIINER, GLiVNCY, LEVY. 21 



PERTURBATIONS OF THE THIRD COORDINATE. 



The perturbations of the third coordinate are derived from Tables LIV, LVi, LVn or D, 

 E„ Ej. The first of tiiese is of the same form a-s tlio tables for (ndz — ljiSz]) and v. After mak- 

 ing analogous transformations and multiplying by the factor i cos i, (i is defined by equation (6)), 



t cos il Up.g iffj'i sin A = lie ^in (x + K) (39) 



Both Table LVi or E, and Table LVu or Ej lead to a single numerical quantity, since all 

 the factors and arguments are known constants. 

 The perturbation u is given by 



u = I cos i [I Up.q r;P)j'« siu A + rt^t { A'l (cos £ — ej + K^ sin s} +c\ (cos £ — €,)+ c^ sin e] (40) 



in which c,, c^, the constants of integration, have not been determined. 

 The constants i\ and c^ are determined by Hansen's conditions: 



w = 01 

 ^w^J = (41) 



ds 



Substituting these relations and equation (39) in equation (40), the determination of c, and 

 c, is given by the solution of 



d'Y 

 C, (cos s — e,) + 6j sin s= —Ik sin (x + A") ; C, sin j — C, cos £ = lJc j- cos (x + K) (42) 



(43) 



where C, = i cos i.Cj 



Cj = t cos i.c, 

 and 



where 



de~l+i(A,' + B,')2V'^ 2 d^ ) ^^*^ 



A double notation is used here, for cos i is the cosine of the inclination of the orbit, and ^ is 



the numerical coefficient of £ in the argument x, but this should cause no confusion. 

 Dividing and multiplying the factor 



by 365.25 



. I cos i-rij ™ -._, 



t cos %-nji= , T (45) 



where Tis the interval in Juhan years, measured from the date of osculation. 

 It is evident that 



. , . Ci (cos £ — «,) + C^ sin e 



can be incorporated m 



lie sin (x + K) 



in the same manner as similar terms were treated in {7id2 — [ndz\). 

 For symmetry of form, let 



I cos i-n^t{Ki (cos £ — ei) + Ajsin E)=Ik' cos ix+ K') (46) 



Finally, then, without change of notation, 



w = Jt sin (x + Z) + Tlk' cos (x + K') (47) 



in which the constants of integration are absorbed in the first term. The perturbation u is 

 tabulated on page 27. 



The perturbations in the heliocentric; coordinates are computed from equations (3) The 

 signs of cos a, cos h, cos c are determined as follows: 



cosa>Oif 0<S2 <180° 

 cos 6 <0 if -00°<Q<+90° 

 cos 6 < in any case, if £ >i 

 cos c > if sin i cos Ji < cos % 

 cos c>0 in any case if ■i<45° 



