Ritter.] l^t> [March 17, 



The coefficients of both these factors are the La Phace coeflficientB, and 

 their values have been tabuUited. Thus the part of the work ruhitlng to 



the determination of expressions for ( ^ ), ( — ), etc., is rendered 



comparatively short and simple. 



In finding J- in terms of the radii vectores of the two bodies and of the 

 cosine of the angle between these radii-vectores, the true anomaly of both 

 bodies is introduced. In the analysis we use the equivalent functions of 

 the eccentric anomaly for those of the true anomaly, and then, wlien 

 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 found the expressions of / '^ V ( " ) , etc., in series, in which 



the angles are the mean anomaly of the disturbed and the eccentric 

 anomaly of the disturbing body, the series are changed at once into others 

 in which both angles are mean anomalies. To efl'ect this transformation 

 there is need of functions called the J functions ; and a chapter is given in 

 which the expressions for these functions are found in a lorm convenient 

 for application. 



When we have the powers of the reciprocal of the distance between 

 the disturbed and disturbing bodies, we next find the term expressing the 

 effect of the action of the disturbing body on the sun. Tliis is effected 

 without difficulty. 



The expressions for the perturbing function and the perturbing forces 

 can now be formed. Instead of using the force involving the true anomaly, 

 the transformation of this, in which the mean anomaly appears instead of 

 the true, has been used. This is the method given by Hansen in his post- 

 humous memoir, in which he lias abandoned some of his former notions. 

 The disturbing forces employed are those in the direction of the disturbed 

 radius-vector, in tlie direction perpendicular to this radius vector, and in 

 the direction perpendicular to tiie plane of the orbit. The forces in these 

 three directions have been deduced from those in the direction of the 



three rectangular axes. The force a ' °° is found at once from the per- 



dg 



turbing function by differentiating with respect to the mean anomaly, g. 



To find the other two forces symbolized by a r. -^ , and o" ^°°, z being 



dr dz 



the coordinate perpendicular to the plane of the orbit, it is necessary to 



multiply a number of series together, two and two, by the formula; of 



plane trigonometry. 



Having the values of the forces, we next find the value of a function 



TF obtained by the integration of the expression 



n. at dg dr 



