J 



Schubert- 3 investigation of KejAer^s Prollem. 



11 



need only io take tUe differential of (a) oiice more, whence we shall 



get 



dH-l.(<f,)r. 



which is 

 duced fro 



inerallv 



educed from r+i, in the same manner as d*^ . (9)" is de- 

 k V ; whence it follows, that the supposed form [a) is 





If in the series (11) we put successively 



1 + 3 



and r=l. K=i-Ht aud r=3, and 



sign +, since no differentials hut those o( an odd decree r are ne"*- 

 ative: whence the equation (11) becomes 



^=ii+ix . (^)^+^+ 



a 



.2 



1.2 



. i (i+3) . (^)'-+^+ 



a; 



.3 



1 ^2 's 



1 * «rsl3 



. i(i-+4) (f-f 5).(^)'+=-f etc. 



Substituting; in this equation the former values (^ 6) 



M=^ (a) 



1 



a? 



1 



2» 



r, <5=f (a) 



1 



a 



I 



2' 



. a? 



c', and A'=: g' ^J,, vvc sliall obtain 



e \t 



(l2)A'•=(^)^[l+<f)> 



.^ex2 j'(i-f5) ^fx4 



I . 2 



•VV •*"•••+ 



,•(,•4.A'4-l^(;-j.A^+2)...(J■4-2A'-n ^e 2k 



1 . 2 



• • # « 



A- 



2V 



§ 11. lu order to develop the series (A) (§ 3) as far as the 

 Ifth power of e, we want the values of x^ and sin ze as far as the 



same power, observing, that since sin it 

 these quantities need only be developed ; 

 Thus, the equation (12) produces the foil 



poiver e^^\ 



2 



e 



tA=:.[l+ + + 





.T5 



7 7 



9 



2 

 3 



s 



3 



A 



5«* 



re« 



£ 



21 e8 



3r,.fio 



4 [1+ 2 + -TT + -^ +-^i^ + ^^^if-] 



** 



A 



I 



[1+ 



3e» 



"IT 



9e 



+ - 



2 





2 



2 



3 ^ e 



4 2 





02 



3.5.1 Kp-'* -, 



v 



2 , e' 



5 2-*.o 



hD+ 



5e* 5e* 



I 



3.52,p« 



2 

 6 



A 



s 



2^S 



[>+ 



Se3 27.e'* 

 "2 ' 2 





06 - L ' 



7 



2« ; 



7^2 





2 



8 



A 



8 



e 



2 



8 



n-D+^'+ 



ll.e* 

 4 





9 



10 



2* .9 



[H-?^]= ,^vo=^';;Lr,+£i]. 4... 



« 



11 



3 



4 



10 



2 



11 



2*^.11' 12^ 



1 2 



e 



13 



215.;-^ 



