I 



10 



Schiibert'S investigation of Kepler^ s Problem. 



(E) sin i i=sm ?>+ 1 e (sin ( 1 -f i;/^-+sin (^i— z j 5* 



t 



+ 



. il C(2+«) sin (2+f) f*+ (2—0 sin (2— i) ^—2 i sin t>) + 



• ••• 



02 10 



^ 



t 



&' 



+ gn 



1.2...71 



(7i4.i)«-isin(ii+i) .«.+(?i — i)"-isin(n— i) ft) 

 .N J (n— 2+9"-isin (?t--2+0/t6-N, (71— 2— i)«-'sin(w-2-f> 



± N,. ( (ri-2r+i)"-isin (n-2r+i)Ai+ (n-2r-i)«-^sin(«- 2?-i),M,) 



2 . . . 



^ i3. The general term of i? (A) (§3) being = +j\'smUfWQ 



still want (he expression of \' in a series proceeding according to 

 the powers of e, to obtain which we may use the same method of 

 functions. Let 1+V(ile2) be called y ; then we shall have k 



y y 



1 



i+\/ (1 



2 



) 



l-v/(l-e^) ^ 2-1/^ ^j^^^,^j.^^.^ ^ 



e 



C 



e 



y 



+ v 







This equation agrees with (1) [% 4) = t/ + x.<p{y) — a, if we sup- 



pose e 



^> <p{y) 



y 



\, a = a; and the sought function 4^ [y) is=-^ 



which, when found, will give x'==e* 4^ (?/}. Now if we again, for the 



w 



. sake of brevity, Write 9, \<\f, %}/', for f (?^), 4/ (^), etc. we shall have 





or|' 



I 



i* 



j+i 



r, 



i.((p}'^^; whence the equation (6) (§6) will 



be transformed into 



(11) 4'=:W-{-ix.(<?)«+2. 



TO" 



d , {Ofr^ 



1 .2 



^y 



+ 



ro? 



1.2i.3 



di/= 



+4 



etc. 



But4»is=— , ^ 





1 



if 



(?>)* ; therefore 



d.(^)" 



d3 {<pY 



«(^)«-x.<^'=-«(^)«+^;^^ 



fi(;H-l).('?)".<^'=+«C«+l) .(<?)"+^ ; 





n(ll+\) (?7+9) . (^)"+l.«': 



« («+l) (n+2) .(<?)"+■' ; etc. 



All tliese differentials are comprehended in the form 



C«) -g-^j = ± «(fi+i)...(7i+r— l).(^)«-h-, 



upper or lower sign taking place, accorc 



t1 



16 



mber 



odd 



To demonstrate the general truth of this form, we 



