1 T'^^^" 7/ 



1893.] ^^* [Ritter. 



A and B being two factors easily determined. W being kno\yn, the func- 

 tion W is next found by simple mechanical processes, and the perturba- 

 tions of z and of the radius-vector are found at once by the equations 



n. dz = nj. W. dt 



-dW 

 dy 

 X being a particular form for g. 



The symbol d designates the perturbation of the quantity to which it is 

 prefixed. 

 The perturbation of the latitude is found by integrating the equation 



d.^ 



cost _ ^ 2 ^ • 



n.dt — ^- « dz' 

 where C is a factor found in the same manner that A and B were. 



To find n.ds, or the variation of the mean anomaly, two integrations 

 are necessary ; in finding the perturbation of the radius-vector, and of the 

 latitude, one integration is needed for each. 



I'he arbitrary constants introduced by these integrations are so deter- 

 mined that the perturbations become zero for the epoch of the elements. 



In making an application of his formulae, Hansen selected the planet 

 Egeria, whose eccentricity is comparatively small, the angle of eccen- 

 tricity being less than five degrees. In making use of the formulae given 

 in the method here presented the eccentricity is considerably larger. The 

 convergence of the series is, however, all that can be desired. In com- 

 puting the perturbations of those of the minor planets whose eccentrici- 

 ties and inclinations are quite large, it may be necessary to divide the 

 circumference into a larger number of parts. In exceptional cases, such 

 as for Pallas, it may be necessary to divide the circumference into thirty-two 

 parts. In case of the applications made of the present method, sixteen 

 divisions have been used : this is the number employed by Hansen for 

 Egeria. 



When a larger number than sixteen is used, the calculation of the 



values of ( , ), ( ^ ) > ^tc., is longer; the process is, however, the 



same in every case. 



After the perturbations have been found it is necessary to have them 

 in convenient form for the computation of ephemerides, and there has, 

 hence, been added the method employed for doing this. 



The writer has endeavored to present the whole theory in a manner 

 easily comprehended by those having a respectable mathematical educa- 

 tion, and in a compass such that the computations can be performed 

 within a reasonably short time. The endeavor throughout has been to 

 use convenient methods, not to devise new ones. 



