No. 3.] MINOR PLANETS— LEUSCHNER, GLANCY, LEVY. 123 



Differentiating eqs. (130) partially with respect to e, substituting in eqs. (128), evaluating 

 the right-hand sides of eqs. (12S), we have eqs. (131, — I3I3), in which the superscript indicates 

 that only terms of first order in the mass are included, and where the argument «? replaces 

 the argument 0. 



For purposes of calculation, the integrations arc arranged as follows: 



In _ _ „ _. _ 



F= lf,+ IK' + [ irj + ( F/' + W," + W,") 



consider first only W/' + W/'+W/' in the integration of eqs. (131). The integrations will 

 concern only part of the terms of first order in iidz^ + ndz^ + iidz.j. It is shown by v. Zeipel that 

 the integration for all three ranks can be performed conveniently at the same time. Let this 

 part of the function be indicated by enclosing it in ( ). The integral 



(n52,(") + (na^^C)) + (n.523<") 



which is a trigonometric series, is given by Z 80, eq. (135), in which the coeflficientsT.p.g are 

 defined by (130) and are easily derived from Table XX\T!I . The coefficients Lp.g are tabulated 

 in Table XXX. 



The remaining terms of rank one which are of first order only, namely, ndz^^''— {ndz^'^'>), are 

 given by the first of Z 81, eqs. (137), in which TF,, TF,, [W^, can be written by inspection 

 from Tables XVII, XVIII, XIX, XXIIo. The fmiction 'is tabulated in Table XXXI. 



The remaining terms of first order in ndz^_ and ndz^ are given by the sum of Z 82, eqs. (139) 

 and (140). The function 



71^2/') - (nfeO) +n523("- (n^Sjti') 

 is given in Table XXXII. 



These developments complete ndz <" within the limits of the tables, and we next consider 

 71^2 '^'. We shall limit ourselves to functions in which the lowest rank is the first or second. 

 Consequently, Tidz^ contributes nothing. 



m'- 

 Any function of second order in the mass and first rank must contain the factor — r and in 

 •^ _ w^ 



the given F {d, e) this factor occurs only in TF/^'. We have, therefore, by Z 80, eq. (131,), for 



one part of n.dz^''\ indicated by parentheses, 



(7)^2,(2)) = r { (1 - e cos £) F,<^' - [(1 - e cos e) F/'']}d£ 



This function is tabulated in Table XXXIII. 



