1893.1 •*-'^^ [Ritter. 



The transformations must be done with great care, and require a large 

 measure of time. In addition to the tedium arising from extended opera- 

 tions of this kind (which must be generally done in duplicate to insure 

 accuracy), many of the processes in various stages of the work are not 

 easily grasped, and certainty is often only secured by performing the 

 numerical calculations. Thus, then, although the method has been pub- 

 lished for a long time, it has been applied only in a very limited number 

 of cases. Watson, in the Preface to his Theoretical Astronomy, says : 

 "The refined and difRcuit analysis and the laborious calculations involved 

 were such that, even after Hansen's methods were made known, astron- 

 omers still adhered to the method of special perturbations by the variation 

 of constants as developed by Lagrange." 



Hansen seems himself to have felt the force of these drawbacks on his 

 method, as in a posthumous memoir devoted to the larger planets he 

 abandons his peculiar method of treatment and uses that of Lagrange. 



As far as the minor planets are concerned, there is no doubt that Hansen's 

 method, as left by him, is too long and difHcult to be practicable. 



What we need now is some mode of determining general perturbations 

 that is easily applied and sufficiently short to attract the efforts of a larger 

 number of competent computers. Only in this way can the constantly 

 growing material be utilized. The new method of treatment will now be 

 given as briefly as possible. 



If A be the distance between the disturbed and disturbing bodies, 

 Hansen has the equation 



n _ ^ _ "HL 



(~) = { 0-q cos (e'- Q) } ' [l-q,{cost'^ Q)] ' 



forfinding ( — V ( — ), etc. 



Instead of the two factors of the second member, I have used a trans- 

 formation of them given by Hill, and have 



n 



(^y = N'' (iJf.a' -2a cos (£'- 0)) ^ (l + b' 



n 

 -2 b COS (^' + Q)) ^ 



n 



Where (i^a"- 2a cos (s' — d)\ " == ["^6 ^°^ -f b ^'^^ cos («' — Q) 



2 'i 



+ 6^J COS f (s — Q) -(- etc. J, 



_ w 

 and similarly for (l -\. b^ — 2b cos (s'-f- Q)\ ^ . 



