3t^ DE PRODTCTIS EX INFINITIS 



5. 47. Alio infiiper modo concinnum thcorema elici 

 poterit ponendo yzzbct^zzzc ^ manente y\ — ^—gzz:bi 

 atque efficiendo Yt productum exprellionum integralium 

 ^^t = 4 , q"od , quo eueniat, oportet effe [^:^'^:^'.jtx',:t^^j^ 

 i ^c^llb) —I- Hoc vero efficietur capiendo oL — a-h 

 {c-hi)lf'^ ^zizf-^ ( ^ -4- I )^ , ex quo reperietur c -H 

 b^i:izo feu ^m — i— r; quare fumatur ^— — 1-|-« j 

 et ^ — ~i--« , atque fequeus prodibit tiieorema : "~ zi: 



fx^-^dx[i-x^yi''Jx^--^{X^^'-^dx[i-x^)-'£^^ 



§- 4S. Sint nunc omnes exponentes r, Z^, y et Ha- 

 aequales, at ^:i=g z=:>j zz^, quaeranturque cafus quibus pro- 

 dudum ambarum expreflionum fiat =: [c!:^,')JyIII'.) • ^^^ ^^' 

 tem eueniet f. reddatur haecfornia »^,x^^>^^±i«, 



(^I^Im^uo) ^ ^ q"^^ fadores ita exprefli , Tt linguli in lc- 

 quentibus membris quantitate b crefcant. Ponatur iam ^ 

 ^{.Q-^i^bziibh-^-tib.^Q^jil^—bli-^h-^^) et a -i- 

 (y -^- i)^i=:^i:-j-^^, feu «— ^(i-H^~y). Porro 

 .fiat /^-(^-l-i)^:=:^^-f-2^,feu/zz^(i-|-^-^)ettf 

 ^[c-^i^bznby-^-^b (eu ^— ^(i -|-y — ^r). Deni- 

 que debebit eflfe a~/ et i?fz=:^, quae duae aequationes 

 requirunt vt fit c — yzn^ — h^ fiue i:-4-^=:y -i-^. Vtt- 

 •de fequens orietur Theorema : ^^~~!^^ 

 fj^i^-^y-^y^dxi r -x^f. /^^^^■^'-'^^-'dxii-xY 



± 



) 



dummodo Gtc-rh 



fxbi^-^^-^h^dx{j-X^)^fx^^^-^'-^^-^dxil-xJ 



j^— y + 0. 



§.49. Alio autem infuper modo exprefTio illa effici 

 poteft zri, ponendo azi:<?4-(r-H i)^ et ^~/-4-{^ 



-i-x) 



