DIFFERENTIALIBVS. 43 



Erit ergo A quantitas conftans arbitraria, a qua fequentes 

 ccefficientes omnes pendent. 



§. 5. Ex his coefficientium \ r aloribus inuentis intelli- 

 gitur, fi ynicus cocfficiens euanuerit, fequentes omnes fimul 

 euanefcere, ita, vt his cafibus valor ipfius v fiat finitus, 

 atque idcirco aequatio affumta 



(a-\-bx n )x 2 ddv-\-(c--\-fx n )xdxdv-\~(g-\-hx n )vdx 2 •==: o 

 integrationem admitat. Si enim fuerit h-\-fm-\-bm(m-i)zro y 

 tum erit v — Ax m ; fin autem fit h-\-j (m-\-n)-\-b(m-\-ri) 

 (m-\-n— 1^— o, tum erit vz=Ax m -\-ISx n -*~ n , atque fi 

 h -\-f(m-\- 2n)-\-b(m-\-2n)(m-\-2n—i )~zo , erit v~Ax m -\- 

 Ex m ~ hn -\- C a' 7 ""^ 2 ". Se mper igitur aequatio propofita integratio- 

 nem admittet, quores fiierit h-\~f\m-\-in)-\-b(m-\-in) 

 (m-\-in — 1 )'___: o; feu hz-z—f(m-\-in) — b(m-\-in) 

 (m-\-in—i) denotante i numerum quemcunque integrum 

 affirmatiuum cyphra non excepta. Interim tamen ii ex- 

 cipiendi iiint cafus quibns denominatores euaneicunt, ita 

 ifta integratio non fuccedit, fi fuerit czzz-a( 2m-\-(i-\-i)n-i) y 

 fi quidem hoc cafu i minor fiierit quam illo. 



§.6". Alter modus ex noftra aequatione valorem ipfius v 

 per feriem eruendi, in hoc conftat, vt ponatur vzzAx* 

 -\- B x k ~ n -\- C x k - 2U -\- D x k ~' n -\- E x k -* n -\- etc. Hinc 

 enim pro v , dv et ddv clebitis valoribus furrogandis re- 

 perietur; h-\-fk-\-bk(k-\)zzzo, quare ponamus h zzz 

 —jk-bk(k-i) Porro vero erit 



J> A(f-+ -cfc -+-ak (k— Q) 



nf-+-nb(ik—n~-i) 



Q t(f->-c(fe— n)-f-n(fe— n) (fe — n— 0) 



/ 2fn-+-:oll(ik— -.v — ») 



J-J ^Lg_-+- e (fr— 7n) -H ?(,' -_-. n ) C„_- 2 T7— Q) 



3 /K_f_ bn(' 2 k — sn— 1) 



F D£g-f-c(fe— 3 n)- +alfc— , n)(/-— 371— 1) V. ~ 



11 — 4./n-4- 4 &» ( 2 fe — +n_T73 etC (£10- 



F 2 



