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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folio 199v, page id 00068196:0394
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Ergo ady 7 20 adq + ax pox. a
Hinc duplicu modo debert
eriri idem nen nempe summando ha fiet a(1) ay+H
20x:a sed idem et obtinebitur et quaesitum hanc
aequationem altimam differentialem cjungendo cum prima
houx ter S Ex 
sib dy: dx = Xx tayaa fiet y = aadu  xxdx,: a x
drllun dydex 2X ddx  dada
dadx
et ay- a d d = 2Xd 4
seu avddx + adydx = act day - XX adx - Xdx dx
sit a dydx cde  xdx di adu = a dq - nta -  ta  - d xdx
eu a exodus - adu dex = dxdx dy t Z Xddx dx3. Sanc aqua
tionem divellamus in duas quarum una sit actydax =dxdxus
Ex priore
ut paulo aute, altera sit aa ddque = 2X dx dx=
fit X  =-logdx. Ergo in proxime praecedente pro Xponendo. log. dx
et praq dx ponendo = a ddx fiet aaddus = - logdx. - a ddx.
seu a ddy = - log dx dux = esx logdx. log seu day = ex logax - dx+ hde
adeoque fiet
.
Hic
11
X: a C
ubi per praecedentia est de = + dx b
ady = dx logdx = dx + = dx bX: a
novam inciditur difficultulem, neque ervim oeccunt alias summanda
sane
Xlogax-
sed video reme
quantitas dx logdx.
adeoque ax logx + Sxdax: de
d, X logdx = de logdx, + X dx. Xddx: ax
loco: Ergua Esce remedsum logax loy (ade)= logax +x: a
avc
praeterca videamus an non liceat modnire logx tollere logax
luy de= log la de) = logdx +x: a
iit prodest
Ergo tandem
ay = Sdx logdx = X +  fdx (2: a
fac l
A ergo Sdx logx = X. l0g (adc) = = xx: a
logdx - 1 + hbX. a = xx + ay: aa = log (ad e) - x: a
+
 ay = X log (a de) =  xx - X + h sax b=ea
EbX: a
eruo habebus duas aequatiopes quarum ope tolli potest loglade) et habebitur
sed nursus aq dy: dx =
ame non videtur spernenda
liber
Haec melhioque non procedit, si divellendo non possit tolli dy
difere
six
tandem qua me thodus
lab
A Re sumamus Methodum Raginae praecedentis nomihulii ma
Et posita aequata pe data dy: de = sext yy,: au
fiat ut ante do y: a= de: devaqe et Yia = d. x: a = dZ: deo. Uende
tatam, et fiat:
quod faciapnns L) logdz: logav et fiet
ddo log du (5) adz logi  Unde, dv logdv. du = dZ logdz - dZ + hdeo. sed compendii
ddv: ddque 3 dodo = d Z. 1Z,: deo dco
causa opmittamus haco seu ponamus h= 0 et erit de: dZ = (7) =  A logdz - 1,: logdo - 1
 per 4 est dco deo +3) dvdv + dZdZ, logdv: log d Z  per 1
= (10) dn dv aa egu Et eodem modo, = (11) dZ dZ ag: XX ope aequationum 7, 10, 11
L ( (per 1,2,) d. X
velutodius Siltollendae do, dZ. Et quidem dvi= dZy: Xinde log dv, L logdZ Flogus - logX. Ergo ex aeq.
+d7 ggdZtloms loyx, ilogaze L y. X
logdz - 1: logelZ + logy = logX= y. X Rursus ex valoribus  ipsius acodeo, 9 et  aequatis
tollamus d et loyde, et fiet Adq
dz daa: Xx, seu
fiet gy+ XX, log dZ + logy - logX, = log dz. 44. seu log (Z  (10g X = log y) (9y + XX) :, gy t Xxdaa
fiet (d2 aq gus: XX) Li_(Z.2, lo9 eZ floy y- logX 3: logdz
et famlibu habe a Hinc habetur et, deuZ, adeoque et   logdo et du, ergo et
ecum hasebita2 GClatio inter X e, a intervemenlibus tamen ipso.
p1 4 logAZ  XoydZ- X = y legdze log y.
Clogx. Ergo logilz (16 ys logus - ylogx +X,:, x=y. Ergo ex sequatis
logacurmis itrum logt
jam prius erat
xaloribus aequationum 15 et 16 habetur  atro idter X et y
Gereraliter sit dy: da = suisions aas ihi a =. C2) luo: deg, et  iio riur e alitua
ddc 0. fiet dus: dx  ddv: adz; C
 loydz: logdo  adeoque hoc
Exgu clav: ddz gieo faciamus a  lag az:
dav: ddque sint seu
o aq. legazue = adv logdo,
() de logdo = de nde Cav: dque = (10) logdz. 1, : logar - 1
tam id agitur ut sublatis ded, de, dZe redeamus ad y et X Fait autem 1XZ.I.
et summando utrobique, fiet dque lvgdz dz
nun jeur ex 2 et Z est 
Erprs t 
s
 l
gjius gles pro
per 10, en
1Z-1,  l0g dv- 1 = (12) Y
qiiglorem ipsius lu
sab
A
quidia gix
atilia isto
dut
jam ex n est ax
10gdZ Flogus- lga 
P.X et
o s. ,eba 3i Ba . 1) itr 7
et