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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 3 (shelfmark LH 13, 3)
# Folio 115v, page id 00068200:0232
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+CXNX
8
itaque cum sit X2 aequ ax + bXX H k. erit _ aequ X. 2. erit
2aequ. a + bX + CXX ete. a vocetur h, et bx + CxX erc sit 2 XX. Erit
2 aequ t + X.X. et X2 aequ hX + XX. X et dx2 aequ. h + (ZX. X + XX. d X
seu d x2 aequ n + (2 bX + (3)C XX +ie scilcet x nqub + CXX+ erc.
10)
Compeniii causa (2) X + X. X vocetur 2d, erit 2 d aequ 2) b + (3) CX + et (.
Et d X2 erit n + X 2d (X2 - xd. X2 aequ h. X F XX X - hx (3 XX. X. H
Ergo X2 - Xd X2 aequ. - X. dX + X aqu ax + bxX + ex)
p Hoc pro calculi pbatione XX. X2,
X dxequ. ax (ebXX (3) C X) Verum)
aequ
=Xdx aequ.
XXX
pro 24
hc. X + XX. X podxq
tea +X. X4
=CX
=bXF - CX3
Ergo pro xeponendo h. X+ XX. X. pro d X2 poncudo tr _ X. 2d
pro X24
h + X. 24d
et ita in reliquis, tunc ex aequ. 5. fiet:
hdx +hdy +X. 2d. dx F 9. 3d. dy + h. V. dx +h. X. dy +h A a ati q 43. dys
+B.H de 1h35. Xdu  gdx k h 2 Z ay
+ yyg. y. L1. de Fapr. B. H dy + 4q 242d. dx th. 27. 13 d. dys
+X. 24d. 742ux,y34421 5i 7 1 17. ad de L. X Z d dy
+X X2G Mg+ F, 9. HBqe dys ryyhi. 243 X xhi Z Iddy
AXXaque gud. Ly3.9t  yp. ). Z  tXXX. X 4 2d sy.
In hac aequatione 20 accurati considerata multae deprehenduntur harmoniae; sub terminis duplicibus
ut 43, subin elligendi similes, ubi 1413, vel 2423, ve (3433; et ila porro
enda
pilantest
Hac aequ. 20 comparanda datae 21. cujus Termun
3. dx; 2. dy; Z. X. dx. Z. y. dy; Z. X. dys; Z. y. dx. Numero 12, ut aute.
tu aequ. 20. Et si quidem comparatio succedit, tanto res facilior. Sin
minus; ex aequ 20 et 21 fiat aequati valores dx aequentur ex aequavibus
dus
20 et 21. unde habebitur aequatio 22, quae erit ordinaria, seu sine
differentialibus, equnda haec aequatio 22 comparanda jam est
cum aequ. assumtitia 2.
Sed compendii majo calculi contracendicausa credo satius fuisset, primum aequationem
5 (in qua omissis redaplicationibus et subintellectis sunt tantum termini decem
Quantitates, equare cum quam aequatione 2I data, quantilatum duodecim
et tum demum instituere explicationem ut fiat aequ. 23. quae explicatis
X2. vel dx2 et similibus, ubi opus, dabit aequ. 24. comparandam cum aequ. 2.
assumtitia. Si comparatio non succedit, demonstratum est, curvam
quaesitam non esse algebraicam. Tunc ad hibendi termini transcendentes
nempe
X. V. et similes. sumatur tatis formula quantumices
generalis, et videatur qum quaenam esse possit, ut ejus differentialis
2 +K xyde
axx+ bxy + cyy
in qua nonobservatur lex homegeneorum ssi abfuisset hx ay fuisset
+6Z+C. Ergo nunc erit dus aequ
aequ. dus aequ. ae
222F2+d
seu log. y. aequ.
Ca X obZAC
et2+52+9
+haycu
dZ + h 97, hxx posito Z aequa.
multae alior non-homogeneae +... et ra
poterunt solvi.
+hyy dus
ay
H ac meltodo
video ex his duci.  sit . qu. exx + sxy tgny x + hX dy, quae est aequatio itque et ex aequ. 1. erit
St algebraica.
+Z+
ta
(20)
equ. 0