

Hoyal Astronomical Society. 381 





dt 2 ' r 03 



4-3^(r ^-). 



" The first of these equations gives r <Hr, and this substituted in 

 the others gives £, 77, £. 



" These are essentially the first equations of the Mecanique 

 Celeste. 



" A second step due to Dr. Brunnow is, that he has executed the 

 numerical calculations of all the multiplications of the prescribed 

 infinite series. This multiplication admits of being made with con- 

 venience and accuracy ; and strictly affords the advantage of being 

 far shorter than the substitutions would be if the numerical values 

 for c were determined by multiplications executed analytically. 

 M. Hansen had already recommended this mode of executing the 

 numerical calculation without having further recourse to any ana- 

 lytical process ; but it has not, so far as I know, been so generally 

 applied in practice. 



"These are the fundamental principles of the solution. Dr. 

 Briinnow, however, has not executed the calculations absolutely in 

 this form, but has introduced the polar coordinates r° and u°, con- 

 formably to his method of treating the subject of special perturba- 

 tions. It is quite possible, since the solution of the equations is 

 thereby considerably simplified (r°dr being already obtained by the 

 first equation, and consequently there only remaining du and 2 to be 

 determined), that the abbreviation is so considerable as to outweigh 

 the disadvantages attending the somewhat less elegant symmetry of 

 the final equations. In the meanwhile, being curious to know to 

 what extent the expressions for £, 77, £ would be found to be less 

 convergent than du, I have also begun the calculation by the formula 

 here given, and hope to be soon prepared with it ; should the dif- 

 ference be not too great, the simple mode of deducing the formulae 

 for £, ?/ and £ will always captivate me. However, this is a mere 

 matter of taste. 



" We have both found the expression for the disturbing force by 



the actual development of — (p being the distance between the two 



planets), and then by multiplication have deduced the values of the 

 resolved forces. We thereby derive the advantage of having fewer 

 values of p 3 to calculate throughout the entire extent of the peri- 

 phery. Perhaps it would be better to develope — into a series 



P 



