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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folio 21r, page id 00068196:0041
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In aequam differentiando pro ae quationibus ad hibeamus formatas
ququande id sufficit, ita inventa in exiguis proproduci poterunt
ad altiora sit formula ahus taam  ejus differentialis aldus taydh taadm
sit formata dadus + (3ydx tudt adidx, ejus multipla per h, potest summari
sit formula asa dus +Vydx tab ux, et multiplicetur per te ejus maltipta
dotam Eiß potest summari, nam fiet ati dy + h)ydx. β +abdx: β
fiet asd h: h = s.V dx: B seu a logh = s. Vdx: (β et  am=  sdhax: 
itaque talis formula a βdus +) ydx + adax multiplicata per h: β potest
sumppari posito tlog. h = sivax: B et summa erit ahy + a (Bhdx: (3. potest
etiam dici esse B
Hinc sequitur formulam
M= (s. ha: B posita b ostante, seu log Rumero cujus lig est 1
P si Parie
cydy + aBdy +) ydx t aD x possesimari
ner posita e constante
seu summari potest  Cydy + aBay + Vyax + ab dx.
ex pmocetenti summari potest
cydy + ahdy +) R ydx + atb ex
A gan
nam
fietimm
Hinc sequitur formulam cydus + atdy + hydx: β +ahldx: 
fietque  Cgy + ahy + a s, Chaxi B
posita  et econstantiou
summan posse, et summa ent _ Lcyy +aly   assx
13
a
+ as dx
1a fdh β
posita h = b. seu bgh = s. Vdxi (3
vidleamus an haec formula summatilis cydus +ahdus ex. conferri possit cum data.
X
ex
a
ero
adp
cydus + at apdus + qydx + ardx. fiet h= p. eergo ams.  = Vax: C ad tih= pae
seu  adeid sed ap p. dx datur, quia datur p ex X, ergo habetur V: β =, a dp:, p. e
e esed hinc ex y ex fit nY: β seu adp = 9 Sed hoc non est in potestate
Et jam video cur summari possit talis aequalio
alitus adus +Jydx bd
 quia constat ex tribus terminis, ex quibus duo multiplicatione talem inter
relationem accipere possunt ut prp productum per se possit summari
Nempe nadue + J’ydx multiplicando per h fit dahdus + n Vh ydx
ebet esse si Vhax= aadh et fit aahdys  aaydh pro nahdy+ Jhydx
sint, sed si sit ydy + Vy ax nulla multiplicatione effici potest
ee
summatio: fiat y =  v firt eB v de + V. V dx seu e. D de +YV x
sed hoc nil puvat, nam sallem termini ordmatae ydus et y deberent habere hanc
selationem ut unum esset alterius summa f. Y Vb fit
rdya yax pertanes YZ  sit adq + haxe, quae multiplicetur per smy nis +
fiet amydy + mhydx dandus tantax sit data aydu ta βdy + Vydx + ab dx
exdus +gygix) compaandoquae multiplicata per m dat amysdus +amβdy +Vmydxtamt ex
et fiel E= B et ar = an= mB et h-Y et m( an
et anh = nB seu rB) = anB sed ita evanescunt
arbitrariae. Et praevideri potest tales summandi modos non procedere, quia
formulis omnibus essept communes non proprie aequationibus curu amet haec
et h = Y
data formula nun sit summabilis, nisi sit aequatio
Ergo
producto
ductum multipticalum per por siet psdus ipsdx dyi  o, quod
hax
ml siplicata pera
quod multiplicetur per  Mdis, et prodibit amdus est fhaxay=
C +amB dx-
quam aequationem advendo priori fret: (tamusdus) t aandus
atae amysdustam Bdy aseis
ma firt amydus + aandus + mhydxtanhax amgaun sar. ay
aq Zdu + GsEdx t ap ydx  p shdx. x (= 0 lint an = m B et aB = Vm - Ih et sißt A M 1 sha
confercadat
fiet. Bate C d= (BR + C βdh + hd B + Vhdx + asfbt - hh x (t  shax sl Y
= axb
aat
h
add licetur per pax. pdx + qdus
fiet et data aequalio mulliplicetur mtq et flic antudus + amhaus + Vmydx f amb dx 1
aq b ux
aq Bdy + Vqy dx
sed prodit difficilior quaesita, quod si sic: au + (tax nu
p- VmtV9-mh
rivide sCcm+
ut fint par aan +9hax = amB+ aq 3. ergo aan = lan s aq - a phalx, et intr + ap = y m + V9fite
Ergo ax amBh + aqBh -  Ihdque + Vm ChaX LV9 shax = mh (hdx = namB tag
ita
ipinitro siti pit
=aa B.) (h a(c. Y) =(1 : )) abi) (aaq = aBh, ) h) quae dificilit
actus t ader seu pro hian potuisend aluti. seu pro sacu i l t sap  BF ette
it
Xoxsy
per PXtqy dat
PXXdy + X ydX