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



COMPARISOX OF TABLES. 



Tables L, LI, LII check satisfactorily. 



Table LIII. — With one exception, the agreement is satisfactory. The bracketed coefficient 

 contains a misprint in sign in v. Zeipel's table. That it is a misprint is evident from Table LVi, 

 in which the correct sign is given to the corresponding coefficient. 



The terms included in the last colrnnn are computed from the additional tables, XIIw^, 

 XIIIio^, 'KlYw' and from first degree terms in Z 116, eq. (200). The latter part, namely, 



[e cos sQ^j(I,-[I,])ch-l Aj[Aj(_v„-[2„])4/.)] 



is added to both eq. (200) and eq. (203). 



Table LIV. — Our table is more extensive. The one bracketed quantity includes an addi- 

 tional term from Table LIII. 



Tables hXi, LVu check satisfactorily. 



CONSTANTS OF INTEGRATION IN n3z AND v. 

 The constants in i were treated in the preceding section by the familiar Hansen method. 



COS I 



It is the purpose of this section to modify the similar treatment of the constants in the per- 

 turbations ndz and v so as to incorporate them in the elements Cj, «„, 7:^, (p^,. Through the con- 

 stants of integration, the constant elements, which have been used from the beginning without 

 definition, are to be explained. 



Since the group method of developing perturbations is built upon the fundamental prin- 

 ciples of Hansen, his conditions for the determination of the constants of integration must be 

 fulfilled. These conditions depend upon the choice of initial osculating or mean elements. 

 Osculating elements are used here. The corresponding conditions are that the perturbations 

 and their first derivatives, at the date of osculation, (< = 0), shall be zero. 



Consider the relation of the constants of integration to the elements. There are two con- 

 stants in each perturbation since the differential equations are of the second order. The con- 

 stant added in the first integration is a velocity; the one added in the second integration is a 

 displacement, or, a perturbation. Now, recalling that the position and velocity of a body for 

 any time t can be transformed into the constants which are ordinarily called the elements of 

 the orbit, it is evident, by analogy, that a displacement of the body and the velocity of the 

 displacement can be transformed similarly into changes in the elements. The four constants 

 in ndz and v are related to the four elements, a, e, n, c, defining the shape and size of the orbit 

 and the position in the orbit, and the two constants in the perturbation which is measured perpen- 

 dicular to the plane of the orbit are related to the elements fi, i, which deternaine the position 

 of the plane of the orbit. It is possible therefore to modify all six elements, but it is v. Zeipel's 

 preference to make the transformations only for the first four constants. 



It is not necessary to compute 



ndz V I 

 dnSz dv\t = 

 ds irs) 



for the following developments perform the transformation analytically, and the changes in 

 the elements can be computed from auxiliary functions. 



Let «„, f„, ::„, c„ be osculating elements; let a, e, tc, c be the osculating elements modified 

 by the constants of integration in the manner indicated above. 



For imdisturbed motion, 



■f„sm£ = c„ + V ,, ._ /1+Cp 



t9(v-7:,)=^\^Jgie 



r cos (i>-;r„)=o. (cos £-f„) r sin (t;-;r„) =a„Vl -«o' sin e 



Hansen's choice of ideal coordinates demands that the coordinates and their velocities 

 shall have the same form of expression for disturbed and undisturbed motion. The ideal polar 



