THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 7 
In the method of this paper we at first employ the mean anomaly of the dis- 
turbed and the eccentric anomaly of the disturbing body, and as soon as we have the 
expressions for the odd powers of the reciprocal of the distance between the bodies, 
we make one transformation so as to have the mean anomalies of both planets in the 
arguments. These angles are retained unchanged throughout the subsequent work, 
enabling us to perform integration at any stage of the work. 
In the expressions for the odd powers of the reciprocal of the distance we have, 
in the present method, the La Place coefficients entering as factors in the coefficients 
of the various arguments. These coefficients have been tabulated by RuNKLE in a 
work published by the Smrrnson1An InstituTIon entitled New Tables for Determin- 
ing the Values of the Coefficients in the Perturbative Function of Planetary Motion ; 
and hence the work relating to the determination of the expressions for the odd powers 
of the reciprocal of the distance is rendered comparatively short and simple. 
In the expression for A’, the square of the distance, the true anomaly is inyolved 
In the analysis we use the equivalent functions of the eccentric anomaly for those of 
the true anomaly, and when making the numerical computations we cause the eccentric 
anomaly of the disturbed body to disappear. This is accomplished by dividing the 
circumference into a certain number of equal parts relative to the mean anomaly and 
employing for the eccentric anomaly its numerical values corresponding to the various 
values of the mean anomaly. 
Having the expressions for the odd powers of the reciprocal of the distance in 
series in which the angles are the mean anomaly of the disturbed body and the 
eccentric anomaly of the disturbing body, we derive, in Chapter II, expressions for 
the J or Besselian functions needed in transforming the series found into others in 
which both the angles will be mean anomalies. 
In Chapter IIT expressions for the determination of the perturbing function and 
the perturbing forces are given. Instead of using the force involving the true anom- 
aly we employ the one involving the mean anomaly. The disturbing forces employed 
are those in the direction of the disturbed radius-vector, in the direction perpendicular 
to this radius-vector, and in the direction perpendicular to the plane of the orbit. 
Having the forces we then find the function W by integrating the expression 
5. Papo 
aw do 
al dr? 
aE -° iG 
in which -A, and B are factors easily determined. 
