SERIERUM SINGULARIUM.
27
1 —y
-! I „-8 (l-r 2OT )(l-r a>W ) ..-3 (l-y- 2m+2 ) (i-r 2 ^) I Ptr
1 W“ 2 1 (l ?/ -2 ’1 fl W~ 4 'l f* '•' -2 ' (' ^ t*
y~~) (1—jr*)
i tum
(i—y 2 ) (!—y 4 ) (i—y G )
•y “ ' v (
usque ad terminum un -{-l
= (1 — y - *) (!+y“ S ) i 1 —y -3 ) ( 1 +y -4 ) • ■ ■ (1 ±y
P]
Quodsi hic pro y accipitur radix propria aequationis y n —1 = 0, puta r, atque
simul statuitur m = n — 1, erit
i—y
i—y"
i—r‘
1 — r~
1—y~
l—y-
i— y
1— r*
1 — r"
r c etc.
usque ad
-y~~ i-
‘in—2
1-y-
ubi notandum, nullum denominatorum 1 — r
aequatio [7] hancce formam assumit
1 — r 4 etc. fieri = 0. Hinc
1 —1— T —I— t 4 —1— ?* 9 —[— etc. -j-A n ’^ 2 = (1—r 4 ) ( 1 —)— v 2 )(1—r 3 ). . . . (1-f-r n + l )
Multiplicando in membro secundo huius aequationis terminum primum per ulti
mum , secundum per penultimum etc., habemus
(1 — r” 1 ) (1 -}- y~ w+1 ) = r — r~ l
__ r n-2_ r -n+2
(1 — r- 3 ) (1 -j- ?*~ ii+3 ) — r 3 —r~ 3
(1 + r~ 4 )(l — r~ n + 4 ) = r n -~ 4 — r~ n + 4 etc.
Ex his productis partialibus facile perspicietur conflari productum
(r — r- 1 ) {r 3 — r~ 3 ) (r 5 —r~ 5 } (r n ~ 4 — r~ n + 4 ) (r n ~ 2 — r~ n + 2 )
quod itaque erit
= 1 -j- r-\-r 4 -\- r 9 -f- etc. -\-r^ n = W
Haec aequatio identica est cum aequ. [5] in art. 12 e progressione prima derivata,
ratiociniaque dein reliqua eodem modo adstruentur, ut in artt. 13 et 14.
4 *