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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folio 367r, page id 00068196:0729
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4 Martii. 68 1093
putabam tali via differentiates primi gradus drnus formari in alpas
EX:a; y:a sit X: a = v. Z
ejusdem gradus constante carentes dy: ex
fiet xia 3 Zdo - Vdz, : ZZ. fiat ergo g: a = (2do -v + 3: coco
(5
us
tutcunque. itaque dy: dx eut  (per 3 et 4) erit  10c0: 22 Itaque tribus
vuloribus ex 2, 45 substitulis in aq. 1. habebitur aequatio cujus  cujus
ope quaerendo valorem ipsius (Zd0- vdz: coco, et eum utrinque aequationem
valais differentiando, habebitur aeqe (B) in qua tantum dv: du per v: Z
)
habebitur. Videbantur haec pulchre procedere sed ecce difficultatem
aut potius obstaculum, ex sola per aequationes2,4, 5, tollantur v:Z 2b
s
Zdv - vaq: coco, habebitur aequatio nova (9) ut qua dabitur dus: dx
per _ x: a et g: adifferens a data quod est contra hypothesin.
ut scilicet aequalio determinativa habeatur, aute adhibitionem
datae unde patet aequationes 2 et 4 non esse pare arbitrarias.
dem est si generalius sic procedamus: dy: x= X. a: y. a
l
H
(3)C
et X. a = D: Z, S2do - vdzue : 10no et _ y: a = v: Z5 (2do- vd5,: c0c0
H
Ergo per
v: ZV
a
ex
Ergo si co = vs2 fiet dy: dx ( ex 2 et 3)= u
(4)
oo
23, 5 tallendo v.2 et (... habetur dy: dx X: a ; y:a differens
non dum licet assumta aequatione data.Itaque si liceat datas
fabricare pro arbitrio poterimus Ron poretongare i finitas, it
at ita utaC sit ipsa 1. poterimus ag. 6 vel 1. redacere ad qua
draturas Nam pu aeq in aeq6, vel 1, Y pro duidx, x. a, y: a
Su
substorendo valores ex aeg. 2 3, 5 habebitur aeq. (7) quam ita
ferentiando praeformataque differentiando ut tollatur summtri
summatoria quantitas habetiur aeque.  (8) in qua solum datitur
Sed si aq. 1. sit ab alio datus difficile erit sic assumere
Valorem ipsius co  relationes in aq. 2 et 3 ut et coimcidat aeg. 6. cum
Cdo: dZ per V. Z.
n6
Redeamus ergo ad hocut ipsam  aequ. 1. nobis
ipsi detnus seu formemus ex D6. sit X. a = v: Z
ipsu aeque 1.
et y: a (ydv- vdz.: vZ ut adeo fitt coco vq
 et g. asv
 ideo ex 2 et 3
fiet dy: dx = Z. v (nam dx: a = Z dv vd Z3: 72 et dy: a = Zlv - raz: v2
s
Ergo dy : dx =  1. 8Z : l: 1Z = 42. vZ = Z. v) Hinc ex aeqe 2 et 5 fiet
c
ne
du: dx = a: X seu dy = dxa: X quae aliunde jam habetur, sed
S flud ergo sumamus exemplum: sit X: a = v: Z, + 10si d vdque : _ vZ + 11
yia  AES2dv - vdque: vE fit dx: da Zdv - vdzue, in, 1: 2Z+. l: vZ
et dy: a = Zdv - vdz in J: vZ. Ergo fit dy, dx = 1: DZ: 1. 32 + 1: vZ = 1: v. Z +1
viZ +1 a jam ex g Ergo ex aeg. 2,3,0
(3)V
fiet X: a = dx: dy -1+ 10y: a + 11 seu dx: du  xa sit 11 - 1 = 12
9
(6
fiet dx: du = X: a - 10 y: a + 12 seu adx + mlus 10 ydy = X dy + nady
Hiae aequatio igitur est resolubilis per quadraturas, seu cont parameter tolli potest.
(0
Nam in aeq. Y. tollantur X: a, y: a, ay.dx per 2, 3, 8, fiet
sed haec aequatio est
cum frnctus sec proinde
V. Z + 10fZetc +11 = v:Z + 10)Zek + 11 
identica, itaque non possiunus tollere n. a, qua, dy. dr C
hac methodo quicquam assequilicet