2J 



=. sill if? 



= sin if? 



wV-2 CO'/. ^io'l^J^^ ^ ^ ^^ 



/15(1+P) 12(H-P) + 64 



105(1 +^-)* 60(1 +P)'+1 20(1 4-F) 60(1+P)f24 8 



oi^k a)V'2 a)^k co'/i 



Etc. Etc. As luis been said, uU the odd differential quotients dis- 

 appear for if?=7i'^' '^'^d ^s *^^ becomes ^^ 1 -\- k" for xp=:'^/,i, 

 we keep : 



'd'P\__ 1 _ 1 ^ 



A/^P 



15 



12 





9 



1—8 A;' 



Vf^if.'Vv.- '^''^' "^''^ '^''' ^'^^ (1+F)V>. {l^k'fl'i (l+/c^)V, 

 For the sake of brevity we have only taken (he part with smx\> 

 into account in the last calculation of the two differential quotients : 



• d\P sin xp 



that with cos i\) is namely =: 0. I.e. of — only the part -^r, 



t/if/-' * i^ It 



and of only the part with sin \]i in the first of the three lines 



d\\}* 



belonging to this. The other parts have every time been necessary 



for the determination of the next higher differential quotient. Proceeding, 



we shonld have found : 



d'F 



225 



360 



136 



-h 



1_88P4-136^^ 



Xl-^k^)"'!^ (1+P)'V2 ' {\J^k"'fhJ (l4-P)V'i 



The coefficients of the highest powers of 1 + k'' are in all these 

 results resp. =: r, (1 X 3)% (1X3x5)% etc. The sum of the 

 coefficients is always =i 1. (9 — 8 = 1 ; 225 — 360 + 136= 1). 

 Hence we get now, taking into consideration that it : ^^ (1 -)- /•■') = 



J) 





sm if? 



=r-=n, and (7^),/,, = 1 : 1- 1+^-^ 



a 



l^l + r sin'if? 



if? (^lf> =: n 



-f 



+ 



2 1-1-^-' 24 



1-8F Q/.Tty 1-88F+136^* (V,^r 



4- etc. 



{l-\k"-y 720 (!+/{;')' 40320 



in which we may also write 1 — n' for 1 : (1 +>(;*) = (a' — -y') : a'. 

 The above series is convei-gent, as is easily seen from the structure 

 of the factors (1 -Sk') : (1 + k^'^ = 9 : (1 + k^'-^ = 8 (1 + k^l^, etc. 



