No. 3.] MINOR PLANETS— LEUSCHNER, GLANCY, LEVY. 121 



complicated that the origin of certain discrepancies is obscure. Aside from possible errors of 

 calculation, differences are duo to the independent adoption of the highest powers of m', w, ri, tj', f, 

 and the number of arguments in a given series or product of series. In most cases our series 

 are more complete than v. Zeipel's. Whether or not the extension of the tables increases the 

 accuracy of the result remains to be seen from future applications of the theory- 

 Tables II-XV. — The discrepancies seem to be due to v. Zeipel's errors of calculation and to 

 their subsequent effects. The larger number of these errors have been traced in the manuscript. 

 Tables XVI, XVII check satisfactorily. 



Table X\^II. — The bracketed quantities in the first three columns are in error through 

 previous discrepancies. We did not discover the som'ce of the general disagreement in terms 

 of the tliird degree, second order in the mass. These terms do not affect v. Zeipel's subsequent 

 tables, since they are of order higher than have been included. 

 Tables XIX, XX agree satisfactorily. 



Table XXI. — The discrepancies are numerous and their origin is obscure because of the 

 very long computation involved. In addition to performing a complete duphcate computation, 

 the terms of first degree and a part of the computation of second degree terms were checked 

 by the solution of the differential equation in the form given in Z 64. With the exception of 

 three or four single instances, the discrepancies occur in two groups, having the following 

 probable explanations. The neglect of the term 



in Z 6.5, eq. (109), by v. Zeipel accounts for one group of differences. The other group can be 



fairly well explained by an error in the addition of second order terms in +:s <^i to c^, — -^<^,. 



Assuming that for these terms he added — w<i>, and, correcting his table, three discrepancies are 

 removed and two others are improved. 



Table XXII. — Considering the disagreements in Table XXI, Table XXII checks satis- 

 factorily. 



Table XXIII-XX"\T^I. — These tables, like II-XV, are simple in construction, and the 

 discrepancies are due to errors of calculation, or they are the result of previous ones, with the 

 exception that some quantities have different numerical values because they are more inclusive. 

 The latter have been indicated by ( ). 



Table XXVIII. — The discrepancies arise from the quantities in parentheses in Table XX\T!I. 

 The omission of the term depending upon the inclination is justifiable in view of its magnitude. 



Table XXIX. — The discrepancies are niunerous and striking, but, since v. Zeipel does not give 

 the formulae of computation, they remain unexplained. The remark is made (Z 77), '"Die 

 Berechnung der Funktion [(1 — e cos s) ( IF3— W3")], welche eine sehr komphcirte war, wird hier 

 nicht im Einzelnen mitgetheilt." For this reason the development of the formulae which we 

 used has been included and the auxiliary functions 2[rj, W3, [(1— e cos e) W/'] have been 

 tabulated. The differences are not serious because of the high rank of the function. Our 

 table is deficient in certain terms whose computation would be long and the omission of which 

 is justifiable in view of their magnitude. 



PERTURBATIONS OF THE MEAN ANOMALY. 



For clearness some of v. Zeipel's developments will be amplified and repeated in an order 

 which we found more convenient. 



The determination of the disturbed mean anomaly is accomplished with the integration 

 of Z 9, eq. (47), (which implicitly contains Z 8, eq. (38)). By Z 9, eq. (43), 



= i{£-e sin £)-g' = ig-g' 



The differential equation is repeated in Z 78, eq. (124), in which is emphasized the fact that 



dW 

 the arguments are functions of both £ and ^, as is the case for —j-' 



