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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 2c
# Folio 94r, page id 00068199:0187
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1ea + uy =0 differentiand sit diea + ydu + 11 dy L0. qram dividendo per H. prodibit. co+ y du + du
esta data diferentalis ejusdem formae, quae sit resalienda  am dx + nydx + aacy ifiet du.u  ndu
etoderc seu Sduiu = Sndxet Rio = StImax et Ga sumax : 1 = yC
M ma + ny =0. ubi pono m et En dari ex x differentiando fiet: adm + dy + yan + ndy-
vel divisis omnibus per n, fiet a dm. +y dn 4 dy =0. sit, jam data differentialis ejusdem formae, qua
netiam bC dantur per
resolvere oporteat:  dx +Cy ax + dy =0 (fiet Sdn: n = Sedx: aa et m = Sbaxiaa sbnax: aa
Ascendamus altius, et esto maxtanus + pyy =0. et diffeientiando aaam + aydn rgydp + andys + 2pydy = 0
Esta, differentialis data ejusdem formae aabdx + acyax + eygex fdy + agyay = 0. divis illa
aequationem per 2p, honc per ag, coincidentiemus, et fiet, an: 2p = af: 9 seu n: p = 25: g, et rursus
ap: 2p = e dx: ag. Et datur p. Ergo etiam datur n. sed fieret denud. a dn: 2p = cdx . g. jam p = gn: 2C
et fiet ca afdn: in = cdx: 9 seu dn: n = ccx: af.. Quod non procedi, nisi in casu speciali
ubi ea ost relatio inter data per quam duo ipsius n valores coincidere possint. Sumammus ergo
aliam differentialem resolvendam bax + C vdx + C vvdx + fdo + v do = c. Et esto v = y + F, et
prodibits + bdx +9 7
ex
+et ax Eyd
e Etax 28 Fydx + e yydx
+Fd+
+ + dt +ydt
F5dy)
+5dy
+ + dy + ydy
audm + a gdn tgy de tandy +ydy 20
2PS
Ex yydx fit do: p =  2 edx: aa. itaque datur P
Exdus fit an: = =F 2f P + 2pt, et a dn = 2f Ht) p + 2p (a s+ d)
sed exyaxsit adn = 2e px + 4 eptc +rpat. Quos duos ipsius
adn valores aequando fit 2aa fap p raat p fraavater aapatm
= 2a cpdx + 4 eptaxA 2a a pat) seu a
quia autem f habetur per X, ideo possumus facere ads = k ax, et aequando duos valores ipsius ap: p fiet
2ef + 2et, = ac + 2C+ = ak. Sed ita infeliciter evanescit t.
AAliter tentemus: adm + ydn + ndy. multiplicemus per aq+ py, proditat:
aatam + aty dn + pyyan + agndus +npydys
+ ap yam
de dr f. efdx- ad df
hanc dividamus per np, et coincidentiemus
ipsi. aequationi O
a+: p = f+t an: n = eax: ua. Ergo datur n.
Compendio in aeq. C terminus infimus sit Saxet peninfimus Ex fiet am = N np: aaque, et rursus an-
npp se dx = npg 7 dx = aagqudn, adeque fier
dn: n = p4 = dx = ApP SVAX,: aa qq = edx: ad seu p9) - pp5 = 9ge, seu
Inp dx = aagdn,: aap. seu
F Sea 43dx, (od 4. P = 5+ t. Ergo explicatione facta redit difficultas prior, secundum +
Assumamus maa + ang + p +  a5 dy: y =0 et differentiando prodit aaam +aydnt andys + dy: 7
eax op nec lucramur
ita frt aayam + agy an t dys  anydus dato simili comparanm
mbyax +cdu summ abile reddi pt multiplicando per in, nam fiet
bmydx +emdus comparandim huic dm yan+ ndus nam fiit n = cm et da =t
et du = Cam + mdc = bm. Eigo prodit dm: m = b - dc,: C. adeoque datur m et. n
Ergo summari  byax +Cdus, m, posito dm: m = b -dc; c entque summa Cmus
(5
fbys dx + Cclus) compendio vocetuo Si ergo summari potst potest srsi = emus
md N = d scmus) d(sm) = Nam t mds datur (m da = datur Et fiet,
et fit se = fbydx + cdys ,: m) ((d Cemus) am) = sbydx +scdu
msi fdim + emus seu N =  SNam + cy
dse = dCemy): m
potitus, am: m = b - dc: c
Sed haec methodus non procedit ad plura simul summanda, ut, si esset
bydx + edus + egydx+ fydy. non licet una eademque multiplicatione efficere, ut
ut ambae partes summabiles reddantur
ad 1042) +  Z db + Chy ryde  p det fdy tyds
in summabilia plura
sed videndum an non liceat divellere, C
2, hedb + eydc
+ f ydb +f yydc
37f bde 4bby df
cyde tegydf
+ ecdus
-
xase ydx, aam textang ax tpgnax t angulus
uit ia
fiea  4fdc ged5 +4d ef dx
= apec dx  apbF Ax
et ase db A = bde + qubeax = Am ec dx tcmbsax
ila habentu
+fc ydy
+bsdy tef ydus
+ g besx + gbsyex + gesqycx
si 2 =1 et 32=
 ita habitil
std aquatio
sin dan ecdxe an cfdx-q
bfax L qedx
gg = apecax a apb f ex- 4495 dc + uaqc d
tos +  et b + Cy
si 3=-E evanescet ydy.
+ e + 57d
Habemus ita quinue arbitrariis
b, c, e, f, g. Cum pim initro
adassent solum tres
fa db + cdy +y de + f ydf + fay + gdx = 0
a
Bovius sic
ab +cy
et fiar
1a db + aa ydc
 abydf
ad + egyax +cgy df
faacdu
+abfdys
+ ef 
sit jam data aequato differenlialis
aax +arydx tpyydx + aagdus + aaydus = 0
.Et datur
firit, df: f = pdx: aa et datur f. Rurtus aa + ab = 4
ab rabgdx: cf = mdx fit g = ncf. dx - a d b,: abdx
9 = a dc + aabdf - aC fndx,: acdx fiit mecfdx= aecab =
fint abac + aabbas- sbefndx abde t acdb
ir dugbus puosemantbus a et valorilysfit apbcedx. ap bb fax- aaqufbde + aaqbe ds = me fedx- a mbsf ax- aaq esab +aaqu fbde
13
49 = mecdx = amb filx- a ag ed b + aaqude
aa, f + t