Ritter.] 1^4 [March 17, 



On a New Method of Determining the General Perturbations of the Minor 



Planets. 



By W. F. McK. Ritter, of Milton, Pa. 

 {Read before the American Philosophical Society, March 17, 1893.) 



In finding the general perturbations of the minor planets the special 

 difficultj^ arises from the large eccentricity and inclination of these bodies. 

 The methods used in case of the major planets fail when applied to the 

 minor, on account of want of convergence in the series. Astronomers 

 were content, therefore, for a long time, with computing the special per- 

 turbations of these bodies from epoch to epoch. Hansen finally succeeded 

 in eflfecting a solution of the problem, and his work entitled, Aiisein- 

 andersetzung einer Zweckmassigen Methode zur Berechnung der Abso- 

 luten Stoungen der Kleiner Planeten, contains all the formulae necessary 

 in the cases thus far occurring. 



Instead of determining the perturbations of the coordinates, rectangu- 

 lar or polar, or of finding the variations of the elements, as had been 

 done by his predecessors, Hansen, in his mode of treatment, regards the 

 elements as constant, and finds what we may term the perturbation of the 

 time. Thus, in place of the time, he uses a function of the time, which 

 he designates by z ; so that if g^ is the mean anomaly at the epoch, we 

 have the mean anomaly at any time, in the disturbed orbit, given by 

 ^o + TOpS, «Q being the mean daily motion, and being one of the constants. 

 If there were no perturbations we should have gQ-\- n^t, t being the time 

 elapsed since the epoch. 



In effecting his solution of the problem, Hansen does not attempt to 

 give general and complete analytical expressions of the series. Instead, 

 he, at the start, converts the coefiicients into numbers, and multiplies the 

 series together, two and two, by the methods of trigonometry. Thus, 

 although we find, finally, the perturbations as functions of the time, that 

 is, have the general perturbations, yet, in applying the method to differ- 

 ent bodies, we must find the values of all the quantities involved for the 

 particular case under consideration. It would be a great advance if we 

 had at hand complete analytical expressions, of sufficient convergence, as 

 is the case with the larger planets. 



Besides the method of multiplying series together by the methods of 

 trigonometrj', which Hansen calls "Mechanical Multiplication" — a 

 method he was the first to employ — he also adopts different angles with 

 which to express his arguments. Thus at the outstart he uses the eccen- 

 tric anomaly for both bodies. When he has computed the powers of the 

 reciprocal of the distance between the disturbed and disturbing bodies, 

 he transforms from eccentric to mean anomaly in case of the disturbing 

 body. And then, when he has expressions for the perturbing function 

 and the forces, he makes another transformation so as to be able to effect 

 the integrations. 



