Fragmeniorum ex Adversariis continuatio. 501 



= zzddl (I — 2*) -H zdzdl (1 — ^z*) — idz"^ (4 -f- 32*), 



unde si inventum fuerit /, tunc erit s = ^— — ii— ' h- ^— ^^ — —• 



zdz zz 



llla autem aequalio ad difierentialem primi gradus reducitur ponendo t = e-^*'^^ dum erit dt = e-^*'^ udz 

 et ddtz=e-^''^^[dv'dz-^v\>dz^), quibus subslitutis reperilur 



zzdv ( 1 _- 2*) -H zzvvdz (1 — z*) -f- vzdz (l -- 5z*) — rfz (4 -f- 3«*) = 0. 



Statuatur v = — -, erit dv=i ■ . — - — . ~ ' ; quibus substitutis nanciscimur 



« (l — «*) a (1 — «*) 2Z (1 — s^f ' ^ 



^ « (1 — a^) z 



Lkmma. Notetur haec reductio /^'^-^"-'^dz (1 — z") ^— ^ = — ^ fz^^^Hz (1 — 2") *~ S «» integretur 

 a 2 = usque z=l. 



Alia methodds eandem.seriem investigandi. Quaeratur separatim series 



- 1.3,, 1.3.5.7,4 



et f_ i ,, , < 3.5 1.3.5.7.9 s 



Pro priore consideretur formula 



hinc erit 



{i^kkz^) * = l-HlM;,4H_!l|ftVH-l^fcV2-4- etc. 



fdpii^kkz*) ^=fdp-+-^kkfz*dp~^^^ kyz^dp^ etc. 



4 -^ '^ 4.8 

 Nunc fiat f*dp=^fdp, et /Wi) = l/2Vp, et fz^^dp = ^^fz^dp, eiit 



yop 4.4 4.4,8.8 



Ex superiore lemmate habemus / ^ = — — - /n g-, unde fit 



(l-«4)"i "*"*" (1-24)4 



11=4/ 1* ^^'"*^' / 1:=8/ — i:- 



(1 - Z*)4 (1 - 24)4 (1 _ «4)4 (1 —«4)4 , 



zzdz 

 Unde patet sumi debere dp = 3-» consequenter erit 



(1 - 2*)^ 



/zzdz p zzdz 

 ^I T-J 1 



(1 — Z*)* (1 - khz*)* (1 — «4)4 



Pro allera serie t = ~ k~^ -r-T-z *' -*- T-r-^-^rrs ** -*- etc. Consideretur 

 4 4.4.9 4.4.8.0.12 



(l-«;z4) 4_| I 1.5,3- 1. 5.9. 5 '10 



£22 4 4.8 4.8.12 



Fial/2'rfp = -^fz^dp, fz^^dp=—fz^dp etc. Hinc rf|) = j-» unde sequitur 



(1-«V 



