125 



Problema 3. 



§. 6. Propont a cadcm aequatione diffcrentiO'diffcrentiali, 

 ut in praecedente prohîcmate , eam primo in aliarn 

 sihi similem formam transfimdcre, ponendo z=:v(i— n^) , 

 tum vero valorem v pcr seriem infmitam convergcn- 

 tein exprimere. 



S o 1 u t i o. 



Cum sit z z=i V {i — ■ n^) , sumtis logarithmis eiit 

 Izzizlv -\- H (l — n n), hinc dilTerentiando : — — -- — - ^-^— , 

 iinde ditTerentiatione iterata emergit 



83z d'^' ddv dv- 2 ô (i -f- nn) 9n* 



, z zs V "VU (i — nn^- ' 



ciii si addatur qLiadratum piioiis : 



d 2* dv^ ^i ndv d n , 4 é- n* 3 n^^ 



zz 'vv I — nn ' (i — nn)^* 



habebimus 



ddz ddv 4éndvdn ^ 2 Ç2 6 n^ — n* — 03n* 



a ■ -v 1 — nn ' (1 — nn)^ 



Nunc aequatio diffeientio - differentialis proposita ita 

 lepraesentctur : 



i; + X(X-hl)«'-^(l-2(X+l)„=)-!^^(l-n«) = 0, 

 factaque substitutione loco ~ et — -, primum membmm 

 absolutum, a quantitate v immime, ita se habebit: 



sive succinctius 



ii -h X C^ H- 1) n= H- ^;^ (2 ~n'(2X-f 2^4-1)); 



