MINOR PLANETS DISCOVERED BY WATSON— LEUSCHNER. 203 



The special tables for each planet are preceded by the adopted elements, a list of the 

 auxiliary quantities needed in ((imputing a geocentric position, and by the developments of ndz, 

 v, and u/cos i. 



The arrangement of the tables for the different planets has been rendered as uniform as 

 seemed expedient, without too extensive transformation, considering the diversity of the 

 original plans adopted by Eichelberger, Ritter, and the author. The angles throughout 

 are expressed in decimals of a degree. Table I of every planet gives the mean anomaly. Tables 

 II-IV give the nonsecular portion of the perturbations. In the tables these are designated as 

 periodic terms. Table V gives the secular portion of the perturbations. Table VI gives the 

 constants for the equator. For the first group, containing eight planets, the perturbation in 

 the third component is tabulated in the form u/cos i, for all others in the form d^=au/cos i. 

 The unit of the tabular values is printed at the head of each table. 



For (93) Minerva, for which e was kept explicit in the plan adopted for tabulation by 

 Eichelberger, a Table VII is added, giving the reduction of the mean to the eccentric anomaly. 

 This table also contains part of the argument N of Tables II-IV of Minerva. Table I of Minerva 

 contains the other part of this argument. 



Tables for the equation to the center and the logarithm of the radius-vector are not given, 

 as Bauschinger's "Tafeln zur Theoretischen Astronomie" and Tietjen's "Tafel zur Berech- 

 nung tier Wahren Anomalie" answer all requirements. 



The perturbations of the first group, containing eight planets, were developed to the nearest 

 second of arc, while the tables give the perturbations to one decimal of the adopted units 0?001 

 and 0.00001. These tabular values were computed only to the last figure given in the tables. 

 The last figure is, therefore, not exact, but was retained to insure greater exactness of the per- 

 turbations to the nearest 0?001 and 0.00001. The tables, therefore, give the perturbations of the 

 first order well within the originally contemplated limit of one minute of arc-. For the remaining 

 four planets the accuracy is still greater, the perturbations having been developed to the nearest 

 tenth of a second of arc and the values in the tables having been computed to one more decimal 

 than tabulated. The mean anomaly and the constants to the equator have been computed 

 to one and two more places than tabulated to facilitate later correction of the elements. 



ARGUMENTS g AND g> OF THE PERTURBATIONS. 



The perturbations are based on the initially adopted elements of a minor planet and Jupiter. 

 They are corrected only for the finally adopted mean mean motion of the planet. The values of 

 the mean anomaly given in Table I for each planet are based on the finally adopted values of the 

 mean mean motion and of the mean anomaly <7 at the epoch, as deduced by the Method of 

 Least Squares from the differences between the theoretical and observed positions. 



In general the absolute corrections to g and it can not be obtained with accuracy from the 

 Least Squares reduction. When they are large they usually are numerically nearly equal and of 

 opposite sign. The position, however, of the planet in its orbit does not depend so much on 

 the absolute values of Ag a and Jtt as on their sum Ju-\-Jg„, which is of greater accuracy. 



Whatever uncertainty may remain in the tabulated values of g is therefore almost wholly 

 eliminated in the undisturbed positions through the constants for the equator, which are based 

 on the value of z resulting from the Least Squares solution. But, in general, this will not be 

 the case for the perturbations, because their coefficients are based on the initial value of n for 

 each planet. 



Theoretically, for attaining the highest accuracy, the coefficients should have been corrected 

 to correspond to the final value of - (and of the other elements), if the arguments were to be 

 based directly on the g in Table I. But within the accuracy aimed at in these tables it is suffi- 

 cient to correct the values of g in Table I by J-=-—n when they are to serve as arguments, 

 and to use them as they stand when they are to serve as mean anomalies. This correction 

 arises from the consideration that the perturbations depend in part on 7t+g , for which the 

 initial and final values are 



"0 + ^0)0 and - + J7z-\-(g (l )„ + Jg„=~ + l (g ) +Jg + J7T']. 

 89369°— vol 10—11 14 



