No. 3.1 MINOR PLANETS— LEUSCHNER, GLANCY, LEVY. 19 



Th&se elements are constants; they differ from constant osculating elements only by the 

 constants of integration in nSz and v. They are to be used in the same manner as Hansen uses 

 constant osculating elements. 



It is possible, in a similar manner, to absorb the constants of integration in the third coordi- 

 nate in the elements •?„ and J2„, but this transformation will be omitted. 



It is a convenience to the computer to have n, and <•, transformed to mean elements. The 

 last term in equation (21) increases in magnitude, progressively with the time. The computa- 

 tion of this terra of large magnitude may be avoided by modifications of the elements n, and c,. 



The method of transformation can be clearly shown from the example {10) Hygiea, 



By equation (1) 



(6.80497„)£ + (8.3192) [n'8z'] = {(>.H0497„) c,-0.''40(37 t- 14f(3 sin e 



+ (G.80-i97„)n82+ {H.3192)[n'Sz'] ^^^ 



It is evident from equations (1), (21), and (23) that the first tei'm on the right-hand side 

 of equation (32) may be combined with the mean anomaly at the epoch to form a mean mean 

 anomaly, given by ^^ ^ ^_ + (fi.80497„)c, 



Furthermore, the second term on the right-hand side of equation (32) may be combined 

 with nt in equation (1). A mean mean motion is thereby introduced, which is given by 



n, = 7i 1 - 0.'4067 = G36f8566 



Again, the third term on the right-hand side of equation (32) may be combined with a 

 term in {Ti32 — [ndz]) which has the argument s. In the construction of {nS2 — [ndz]) there 

 occurred the terms ^ 3^,3 ^j^ ^ ^ ^,q cos e = (1 .545) sin (e + 7?53) 



The addition of — 14f6 sin e from equation (32) gives 



-|-20f2 sin s-h4r6 cos £=(1.320) sin (£+12?74) 



These two values for the argument x = e are tabulated in the body of the table given on p. 27. 



Further, since it is intended to improve the perturbations by the use of Xewcomb's value 

 for the mass of Jupiter, ndz must be multipUed by the factor 1.00050. The combination of the 

 correction for the mass of Jupiter with the term of the same form in equation (32) gives 



(-F 0.00050 -0.00064)r!.52= -0.00014 ndz 



This correction is the la-st step in the determination of ndz, since it depends upon the pertur- 

 bation itself. 



Without change of notation for ndz, the collected results are: 



£ — e sin s = c2 + ndz + n2t (33) 



where 



ndz = [noz]i + {ndz-[ndz]) -O.OOOli ndz+ (8.319) [n'dz'] (34) 



It must be remembered that [ndz]i and (ndz — [ndz]) are numerically different from their 

 original values, but there should be no confusion if this transformation is not made before the 

 con-stant c has been determined. 



The constant elements are now: 



Epoch and Osculation, 1851, Sept. 17.0, Ber. M. T. 

 Tij = 636.''8566 = 0? 17690461 



