670 SCHÜLTZ-STEINHEIL, THE ARGUMENT Xm- 



Tlius by the above mentioned method we do not efFect the 

 disappearance of the small divisors; but tliis is not our aim in 

 as much more as it is proved that these must for the most part 

 occur in any form. Our chief gain consists in avoiding by this 

 means a very laborious tabulation. 



In »Grunddragen af en metod för beräkningen af absoluta 

 störingar, med hufvudsakligaste afseende på de små planeternas 

 banor» Gyldén bas given formulas for determining the absolute 

 perturbations of a small planet; he has, however, not developed 

 the formulas in detail, or made them fit for numerical calcula- 

 tions, and, so far as I am aware, they have never been practically 

 employed. Gyldén, contrary to Chaelibr, introduces the argument 



Xm already in the developation of — , and he therefore must 



calculate this quantity as well as all following quantities sepa- 

 rately for an odd and an even m, which makes the necessary 

 labour very great in comparison with Hansen's. In other 

 points too he deviates from Hansen in such a way that the 

 calculations seem to become more difficult than necessary. I have 

 therefore thought it better to follow the method given by Hansen 

 in his »Auseinandersetzung etc.» for determining the quantities 



a -TT-, ar^ and a-^TT^ and thus develope them with arguments 

 OE or oZ 



ie — i' V and afterwards change this argument to an argument 



is — i'Xjn . It is my intention in a future paper to evaluate the 



absolute perturbations of a small planet by this method, and I 



will here give the formulas necessary for this calculation. 



In Hansen's »Auseinandersetzung etc.» we find the formules 



necessary for determining the coefficients in the following ex- 



pressions : 



on 



ds 

 ar^= I2c(i i' c) cos {is — i' V) + I2c{i i's) sin {is — i' V) } (I) 



a-jf^= 2Id{i i' c) cos {is — i' V) + ^^d{i i's) sin {is — i' V) 



a-^= 22ß{i i's) cos {is -— i' V) — 21ß{i i' c) sin {is — i' V) \ 



