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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folio 147r, page id 00068196:0291
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assumta
sit equatio   ad curvam Generalis Homoeoptota 2, cum sua differentiata 2. vel 3
O= 100 + WWBX + 120X, + III Xy + 130 F, + 121 Xg, + 140 X+ + 1BIXy, + 122 Xy vtsoX, BXy raexy iili  p
o2. HolX FIB. ZA L AoXHr +, 121 XydX F 1213 dx + 3 130 X dx +, IXy dex 4, 131 Xy dX + 13  dx + 4140 X dx
ydys
rydy xii ydy xydy
rydus xdy
xyt
Pay B (142 Xg + 123Xg G
43 132Xyax +Ze ex +4. 141 Xydx L 141y ex +G150X dx +3. 133 X y ax L 4 14C X ytx r2.192 X u dx. + 5. 15VEydx t 151 dx + 6. 160X d et
Xy dy Xyly Ashay xdy
5
Ndy x xyay grdu nyay xdy y’ay
Hade conjunbrequatio t  Hini ex aeq. q 2. quaerendo
valorem ipsius dy: dx fiet dyrdxL m y
(3) io + myL oXLA EIBIXy +12II +3130X L2 M axy + 3. 131 X y 4 B31 V  q aoX  F 3. exg r2. 32    +412 Xy t 4 1y  5.1X et
x notMIXH oy 42. 121 Xy + 121 X L 3. 130y + 2MXy + 3. 130 Xiy +131 X +4, 140y dy + 3. 132 F j t2. 1 Xy  (t. (41Xy +1 41 X +5. 109 etC
se Huic jam  aequationi 2 vel 3. conjungatur aequatio differentialis, generalis, homoeo-
ptota data, quae redacta ad  simplex, seu in qua dy: ex non ascendat ultra simplicitatel
quae redacta ad valorem sic sabit - e abo + vor vr horyt f ubi notandum in oge dipirqui ubi disse cnliet p b
demntgoXtg + ZIIXG+ Seax qui + 3EXytq12
lequitu partemm sortiorem ut, et xyxadbi inxid
Ly (4) 200 + LVBX + 21y + Zuxy+ 220X + 202y + 2EIKy + 212 Xy + 23OX + 2039 + 222 Xy + 231 Xy + 213Xy + 240 X̀ + 204y et
dx 200 + 201y +210X + 21IXy + 220y + 202X + 221 X 1 + 212 Xy + 230G+ 253X + 222 Xy + 231 Xy + 213 Xyt 290γ 1204X et
Ubi notandum inaequatione F. distingui quoad n ex pressiones coefficientium; nempe
ubi differentialis x vel dy, sequitur partem fortiorem sul in XGdx ) si scilicet oquater 4
dy-
ud nligiem (si scilicet aralogra 4 ad aequationem redureritur) ibi in coefficiente ex
duobus numeris praeponi majorem) hinc coefficiens hujus formae est 221, ubi ex posterioribus duabus
potis inter has tres 2, 2 1 nempe ex 2 et 1. praeponitur 2. major et scribitur 221 X ydx 
Xist dyue.
sed si differentralis sequatur debiliorem ut in Xy dx pai praeponi numerum minorem, sundel
lujus coefficiens est 212 et seribitur 212XY ax 7 Quod si aequales sint X et y adeo at discerni
nequeat quam partem sequatur differentiatis tunc etiam numeri sant aequales (ut in 21 xylx
Kydys
Superest jam ut aequentur inter se valores 2et ex ag. 3 et 4. quod al fial 
felicius et universalias  pro aequatione, 3, prius scribamus aliam generaliorem ad modum
aequationis 4, ne scilicet calculum finstra nobis reduamus difficilem e ac speciulo
dyS) 200 + 3∞X + 3My F3IIXy L 12OX + 302yF G EIXGA912XY 4 330X + 303y 4 3reX Yy 533IXy + 313X9 1340 X + 304  etc
ax 300 + 310Y +301 y +3 11Xy +320y + 302X + 32(XG F3I2Xy +330 y + 303X + 322 Xiy + 331 XY +313Xy + 340 7 4304 X et
) Horum inferius fit ex superoore mutendo 3 in 1 in antepe
ma
Aequando iptur vol ubi 300 310. 30L 3F
nota, et penultimum
papsat genultio
amim
tequo
eatum
ti
q
Ubi 300. 310. 301. 311. 320. 302. 321. 312. 330. 303. 322, 331. 313. 340. 304 el stu
Valent 110 2.120. 111, 2.121. 3. 130. 121. 3.131. 2127. 4,40. 131. 3. 132 4.81. 2. 137. 5. 150. 141, et. multiplicalun, per rotam s ecto pemin e  rus
lus fornis
Aequando igitur valores duos ipsius dy: ax ex 59. 4 et 5. prodibit aeque
duobus invicem ductisnu
indua inersi
200. 390 + _o 3or X 1 201. 300) t ef 3a Xy + 220 500 X1 202 30y4 22130X y 1 et 2 21039
In aeq. 5. haec lex est, 
ut coessicients anus affectus signo t, ulter affectus
0300 + 210. 300X + 211. 300F) F. (unaaffecta simoplus altera suino minus; in priore pearu enultium aota refert potentiam ipsius x ultima ipsius 
32
=200. 304
D duorum nomeroii posteriore cuufficiense contra sixota posterior unique numeri penultima
addituducatur it alterius anterennetimat et viceversa, et sit prodeunt duorum literaru
N - ZM. 300 Xy
Xret 300 
estermina
es. Jiincissine
d
ntum
 majorem
 totus ed Enteo praevider
prduc
+200.300 + 210. 300X = 211 300Xy F 220. 300X + 221300x y + 230.300X, + 222. 300X y + 231. 300 X y L 240. 2aY. 1231300Xy, + 241300X9 + 250300 8 es
F201. 300y
tr02. 300G. 4212. 300 X i +203. 3009
+213. 300714204. 30  f223300X7 F214. 300 X., t2o5. 300 )1,
E2E 310XIL +213. 3 1 X, t2013 0y,
+200. 30 y +210. 310xy +201. 310 f ,2 1BX 4292. 310y 1221. 31Opy + 212. 310 Xg, 1203. 310  1231. 31. X  1240 310 Xy
4230. 310 X. 9
=200. 301X L201. 39) Xy 1210. 3015 f 2u. 301 p ( B , 220. 301X +212. 301X 1f + 221. 301 p 4 +230. 301X, -2. 301 X 1 +231. 3015 a2o 30 X
+220. 310X8
+217 301 X. y
F210. 311 Xy
4201. 3II Xy,
4203301 7
A211. 311 Xy 4220. 3I1Xy
+ 202. 311 x
1200. 36d x + 201. 3027. p . 1210302D 42I. Vgris = 21. 25 gry. tnarp. 2I22B, tey miptus +uorer
2 20. 320y, 1 20. 32o X.  12013 20Y3 +220. 320 Xy LN. 320 X, 1or goy 1 2 i1. 3cor  C.. 1e12. 2dop. ) tor Mo 
In hac jam aequabione tolli
+200. 321Xy H, = 20.321 X +301. 321 81y
possunt omnes formae, sese eodem a200. 312Xy
modo habentes ad X et y, et ut
reliquum ibitipotest per A9
=201. 312X7 1210. 3128 f
+200. 3IXY
=20. 330
24b0.237
+ 260. 322X7
t213. 201 X 9. 1 264. 2t py
+221. 311Xy +230311X 
4+212. 311 X 3 103. 311 Xyt
1203. 392
1230. 320X 
:21. 321 x +202.321Y4
2220. 321 X
=211. 312Xy, =220. 312XY
1202. 312815
1210. 37 0xi 120121013, = 28. 230 X  Fes. Hoxu X 202 30
=cl. 303 (.  2 210. 103F a 3 202. 363 p f. ; 1211. 303 F h 32Eb 233 ,
2 200. 213
+200. 340a
1210, 322
ro
20 3t F. (2di vro. 331 Xyt
L 21. 313 X
1210. 33 X
7210.340Xy Y201340y3
+20. 304  25  F2f1 304 Xy 421. 3047
=200. 213 Xy, = 200. 314 X y X180.305
troo. srais
coneisis ircetsod
= 200. 3317 y
xy, ry, xyt, xis etc. et
es prodibit 69. _i 7