20(5 Indagini 



Di nuovo l'equazione (B') con la fofl:ituzione u' =. — fi 



X 



cangia in quefi-a 



(B"; . . . Mdx" - ' = v"dx"-' [e - 2/(OT -f I ) + 3/, (m 4- I) (w + 2 ) 



— 4/( w-j- i). .,(?«+ 3)4. 5^y(?w+ i)... (m + 4)— ecc. 

 -}-((?- 2^(w -f i) A- iKim V i) (;« +2) ~4/'(w -:- i)...(w + 3) 



+ jr (W4- i) . . . (W4- 4) — ecc. ) J)4- (/- 3/j (w 4- i) 

 -]- 6/(m 4- 1 ) (m 4- 2) — I Qq{m 4- 1) (w + 2 ) (m + 3) 4- ecc. 

 -f(c? — 3K(w-f- I )4.6/'(?M-f- I )( w-}- 2 ) 



— I Oriana 4- i)...(w 4-3)4- ecc.) j) {xdv"dx''-^ - v'dx"-') 

 1 -' ~^i^ — --^l (m^ i )~^ loq (m~{-i)(m-\-2) — ecc. 



'- • +(K — ^p(m~\~i)-\~ior(m-\-i)(m+2) — (icc.')Xy 



- (^x'ddv"dx"-' — 2xdv"dx'"-'^2v"dx''~^y 



H~ecc. 

 dalla quale combinandone fimilmente tre, come fegue, 

 ÌB'') . . . Mdx"-' = xdv'dx"-' (/- 3-5(w 4- 1) 4- 6l(m + i) (m j- 2). 



— I o^ (w 4- 1 ) . . . (/J2 4- 3} 4- ecc. — 2/^ 4- 2 . 4/ (m 4- I ) 



— z.ioq{m-\-i ) {ni -\- 1) -\- Qcc.-\- {g — 3K(w-|- i > 

 -]- 6;'(;'>2 -[- i) (;w _^ 2) — 1 0(7(w -[- I ) . . . (w 4- 3) + ecc. 



— 2K4- 2.:ifp{m-\- i)~z.\or{m-\- \){rn-\- 2)4-ecc.)Z)' 



4" x^ddv"dx''-^ {h — 4/(/w + 1 ) + I o^(;w + i ) (?w + 2) - ecc. 



•\-{K — ^p(m-\- i)-j- ioY{m-{- i)i>n-\- 2) — ecc.)j} 



4- ecc. 

 ( C ) . . . £•_ 2/(?w 4- i) 4- 3^5 (w 4- i) (w 4- i) — 4/('^ 4- i)...(w + 3) 



+ 5'7(^+ i)..'.(?«4-4)— ecc.— /+3/^(m4- i) 



• — 6l(m~\- i)(m-^i)-\- ioq(m'}- i).,.(/»-{-3) — Qcc^ 



+ 2/^—2.4/(^4- I )4-2.io(7(w4- I )(W4- 2) -ecc. =0 



(C') ...£■- 2^(w + i) + ^K(m + 1) (w + 2) - 4/'(m + i) (w + 2) (mi^) 



