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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 2b
# Folio 57r, page id 00068198:0113
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Melius in alia scheda
Aequatio datdy: du (+ my +ngus t prsfek 3:, 9 Hry ttyysy Aet=  equatio quaesita
differentialions liberata sit 0 at bytcgy + eu+ etc quae differentiata duubit
=24. d30 +23, 431 + e2 d32F y Bg lx, da + ydb + yydc + ye etc a 0+ 1bG k29) + 3Cyy + 4f Y+ et =3 -0
ed
1. 15. 31 72. 142 +3,13 33+. 12. 34 +5. 11. 35116 = 25. d30 424. a31 + 23cera l3 3 a  u dx 4 + dyt =0 seu dh: dx = 4: t = + = C: D. seu 2d = tC. seu fiet
Il
lemyangus rete  ibys F2cGt3 eyy ret  otryt lyy tete, da t yssot yyde + yderen
1be+ Edyt 3egys + 4ffy +5gl y 
+Am tem +hem 1+fm
Sit aequatio data - dy: dx = 1 = 10 FH y + 12gy +13y retC, :, 20. + 219 + 23 yyt23y ek = 2 - 0. 7
Sit aequatio quaesit 0= 3 = 30 +, 31y. +32yy +33y + etc. quae differentiata dabit
0= 4 = dx (d30 + y d31 F gy d32 +y d33 t etc) + dy (c31 + 2. 32y+ 3. 33yy +4. 39y t etc
seu 0 = 5 = dx = + dyC unde - dy: dx = 5 = 2 it = 6 = C: D. seu C3t - 4
1. 10. 31) +2. 10. 32Y + 3. 10. 33 yy + 4. 1034y t ete .  20430. + 30d3y + 20 d3 ryy + 2o 43Gre
+1. 11. 31 2. 11. 32
1 3. 11. 33
a 1. 12, 315
v1
2. 12. 32
1. 13. 31
YZ1. 430. 21 431 21432
determinantur
 ubi 10, 11, 12 etc, 20, 21, 22, ete
sit 10=0 et sit 30 constans seu d30 0  fiet:
dx, 11. 11. 31, 
dx; J. 12 31 + 2. 11. 32,
31 + 2. 17. 32 + 3. 11. 33
22d30 22 d31
30, 31, 324 etc d ex X
g
=10 = 20 a3I
=11 = 20 d32 + 21 d 30dx
=12 = 30 33 + (21 d34 + 22 d31) ex
et
etc
23 d30
dX, li. 18. 31 + et c3. 32
Et s e habetur solutio. Nam pe quia generaliter possumus
solvere hanc aquationem. Z+. dque hv, +i dv + K dX o posito h, i, k
dari per X, hinc h solvi possunt aequationes 10, 11, 12, etc. Nam per ag +0
 sit dsts dxi n ex. 14. 20 et habetur per 31 Hinc porro per
aequu sit Nam  in aequ. 7 termini ultimi utrobique evanestunt ob 10 =0, et d30 =0
uati
sed 31 ex = d31 fit  dX 1. 11. 31 = 20 d31 seu dx, 1. 11: 20 = d31: 31 quod
in raquat sed dermini pengise
aequatia
haberi potest per Cogarithmos. Termini an depenultimi aequati, dant:
dx, 1. 12. 31 + 2. 11. 33 = 1 20 432dX + 21 d 31 dx  seu dx, 1.12. 31 +2. 1. 32 = 11 = 20 d32+ 21 d Vd
se ubi eum 31 jam habeatur fiet +1. 12 3,1 edx + 2d. 32 dx = 20 d32 quod perinde est
ac si pro 32 ponendo V, et pro Z. I1 ponendo h, et pro 1. 13. 31 - 1. 21 431 ponendo K et pro-20
similiter inventis jam
1. 21. 431
a Kdx +hodx F idv ∞0
31 et 32 inveniri potest 33 per aequ. 12. simili methodo. Nam aequandi
ponendo;  scripsissemus
comparando Terminos ter quartos perantepenultimos, fiet q 11. 3F +E
dx) (1. 1. 71. + 2. 12. 32 + 3. 11. 33 = 22 d3i dx + 2 d3e  +, 20 433de C et pro od. 33 dX
ponendo  d33, fiet fol. 13. 31 + 2. 12. 32) dx - H 3. 11. 33dx f 2- 20 d33 =0
∞1. 22d3100. 21 432
quae ide ope pujus 33 habeni potest ad modum Aequationis. 13
 Itaque  ad omnes Aequationes Tancentiales rationihes solve
membra.
determinandas tantum dispiciendum an possint. tolli terini  in quibus
est  qua per unam tantum ordinatam afficiuntur quae
comprehenduntur hoc membro: dx t Idx posito ID da determinari
sid
quomodocunque non  per non aliam indeterminatam quam a
a sit A
X = 200 + 201Z:, 30021301
Et ubi 100 101, 102, etc sunt intre sed 200, 201, 300 30l ad hibentur ad tollendum, terminum 
ergo sit 10 = 100 + 101 X + 182 XX + 103 X + etc
terminum 93. quanquae 301 possit poni 0. ita in aeq. du aequatione data seu 1, quae
erat 102 -dus3 dx = 10 4 1y+yy + etC 7: 120 + (21 y +22yy+ et C. in 
in 10 de primetur X   ab X ad X Z  a 7) ad 2Z. Sed eodem modo faciendo
pur
 et p destruetur  ubi rursus
F.8 +.. 8 et faciendo =. 600  + 601: 300l 470
rursus Z = 400 + 401, :, 5008+501; destrub
at selal inbetiur soiauta ritius  srmentire cotantur mil prues seruagus et 13)