r 



6 Schiiberfs investigation of Kefler^s Problem, 



If we substitute these values ia the equatioa (3), we shall obtain 





i.'.<P l 



0'=:+r.(^)H2^V^'; 



V' QpY—Q r- {4) ^ . ^'— 6^'.> . (^') * — ^Af. C^) ^ . 9"', 



da- 



l=+f"'.(^)*+l2^^"^(^)^p'+36.^".(^)».(^')'+l2^^".(.f)3.(^"+24^'.^.(^')' 



+ 36 -,/.'. (^)«. ^'. ^"+ 4 ^'' {ip) 3. 4."'^ and so on. 



But it is evident, by the conimou rules of the calculus differ- 

 entialis, that these expressions are susceptible of a much more sim- 

 pie form, thus : - 



{^) 





^Vj 



dd^ 



d^a 



d.^\(<g)\ d34^ 





J A — + — J , and so on. 

 ax* dys 



By substituting these values in the equation (2) (§ 5), we obtain 



(6)^ 



u 



^ 



JC 



8 



^ ' (<P) 



XS 



1.2 



dy 



^ ' (<5) 



1. 2. 3. 



dy 



+ etc. 



the letters ^|/', ^, and their differentials denoting the particular 

 value, which these functions have when x 



whence 



y 



a,u 



^^ 



c 



da > 



<p 



(§ 



§7. T 



h 



le application of this formula to Kepler^s Problem 

 no other difficulty, than the length of the calculation. 



We have only to compare the equation I 

 {i)o 



i. 



e sm i 



with 



SP 



y + 



e, a 



H 



y 



'■i 



9 



3 get 



siu «. 



Let, in the first place, the sought function be e (§ 4) : then we 

 shall have 4/ [y) 

 f(a) 



y=i 



.X 



f> 



dy 



dy 



i 



d the equation (6) will be transformed 



(B) 



i 



>t + e sin .« + 



e^ d.sin*^ 



1.2 



+ + 



€n 



d^^^.sin^f^ 



1.2.„n ' d g^n^i 



\ 



V 



