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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 2b
# Folio 73r, page id 00068198:0145
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1
Videamus ax remedium comminisci liceat novam arbitrariam insecendo
qualis est d occurrit discendendo ad differentio differentiales
et ponendo constantem differentiam  deo de
Ct
deu dy- Xx + yyrdx ady = 2x dxdx + XXddx F yyddx + rydydx
et pro dyponendo valorem fit ddy = 2x dxax +2yxx dxdx + 2y dxdx + (xxtyy) ddx
sit condx dx dx et fiet co ddx = dxdx, adeoque habebimus
ex +igxxt2 (2C0 (xt yxx + 1) + XXFgy) dax = adu. sit ergo
sa ddx = dz: Z +advro 1 +  d do:v, de
posito de esse constantem,
et fit dx = log d log ZV de tede Rursus 20 (x + yXx + y) + XX +gy = _ ZV
et dy = 7V de + fde Et de
AX
Zu+f
10g  log3u+ F
=XXXFyy
Erpo ZV + f = XXFyy log ZV + C(XXrgy) ubi pro ZV substituendo
valorem habetur valor ipsius  co per X et y seu ipsius dxax: ddx
ubi pro ddx et dx ponendo valores, habetur aequatio in qua logzv,
et dlogzV et de. differentietur et aq. 0 ejus ope habebitur rursus aeq.
in qua ZV X et y, dy et dx et tollendo duidx habebitur aequatio in qua
iterum ZV, et duas illas aequso tollendo dabitur quae situm:
ed iideanius
lo
Beo
ddx= de compondio ∞ =  fiet adx= aet ax Dade + cde
ady =  d 8dlog8 de et y dy = 8de + et dy: dx = 8+5, :, log5 +c
ae
= XXFgy seu + f = XX ryy,  log. e +C
dxdx
superest
d. 2dxdx (X Byxx 1 y3) + ddx (XX ryy) = ddue unde explicando fiet
aoe
2dede l4 qu lloge rC) + d dloges  seu (2dxdx (X tyxxty3) +XX rgy) dax=
da
x
et fiat 2dxdx (XAyx r) +xxtyy = 40 = dele qu (log  F C)x Fyxxt u) tnarnus c X
ddx
Rursus dy c+f
x
Joges +C
dlog80)
Xx+yy. quam aequationem diferentiando
fiet (0gP+ c) des + (DH) dlog 8 +2X (0g8 +C) de
ub
qu (loge+C)
m
+2y ((+f) de
unde fiet d ae = 2 + 7 ( (xlog erC) + y ( 7 5), 2 qu loge +C) D xd
do
l0g+C + 1+ (f.)
x essu losgris
YI =XX - yy
Ita habemus duas aequationes C) et D, quarum ope tollendo  habetur relatio
in inter X et y fundata indes in exponentialibus. vel quod idem est ope harum
duarum aequationum habetur curva p quaesita per intersectionem duarum superficie
rum
pece dum video, quid successum impedire queat, ets. via sit plane extra
pdinaria, et nimium praestare videri possit. Ex laequ. dtollendo log e peraeuiss
habebitum valor ipsius  per X et y, ergo et valor ipsius e loge. t utrumque 
substituendo in aeq. ( habebitur, relatio inter X et y, quae Cujus differentiatio
debet reddere differentiationem differentialem datam, aut oportet ubam quadcatura
opdinariam succedene, aut deuique eut fectus in Methodo. Hoc ergo absol-
1
eendum cum mive sit promissrium.
Generaliter dy. dx = +: a fita ddus = dxdx dt  +dus dx d t+ dxt.
Si faceremrus dax = 0 et pro dus poneremu dita dxt. a fieret audius = +t a + t dt dx dx
fit ita ergit de corstans jam sit aa dau = dg et dx = log x d8. 8, et dl + t dt = 
adueDax dg. ex  g. a e +: a = put finc adque. 8dg. des sugo y-  sed hoc
non licee namifesta t tervenus ergo dex, et aser a ddue = (ax (dxdt + dus db) + +) adX
addus= d de et ldx = d d9 de:c, itady = ∞de a et dx  de log 8t sit
jam + + dxax dyax d+ = 8 ita enim hoc in daq dato odus pro dus, E ax, dax substituatur
sit e yt
valor fit et + de q qu l0g5,+ dxx t Et dede qulog8 t t =+. seu a- at,:, agulog e Al +  quloyp. tdG
laede 4
aqu. de: de
porro ob novos valores ay: ex fit (9+3, de, : a):   de log8, i 129, oblog = +
agde.
Hanc aeq. iteum differentiamus fiet do loc  A_ (19) ds. 9): ( qu loga= duat + dx at, a
Sdergdo,ia deloges
hgo de :de = plog.s (8 19) 1: ); gu log c (18+ 9= ta log e c t)   - p+
seu fiet a log8- 9 - 49: (+9 + a loga dt = 8- t,:, adF
qu loges (a de + tt
fXl +
et _ (a log8+ - 8d+ - ag dt + + 8 lg 9+  ( (9+ 9) + d Cd = (91 + 89) at  (9+9) t+ t
seu (a8 + a9 + at log8) dt =  (8 +9) G dt, 
+aGlogqat- at loga aF
3
seu a (8 + 9) (A F + at) = a t log8 t seu positos . 9 = 0 fiet FGd t + a at = alto
quod debet esse verum universaliter sed credibile non est, it tam generalio negula hanc rem absolum
posse. Et sane hunc tollendo  hanc 54 coogendo prior 49. Ibg_= t. fiet t = 8a) +a
m  odipd habtundum an eoa es ute tet a
sal
 mad oiet incura snto