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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folio 294v, page id 00068196:0584
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c
Onnis aquatio ad curvam sic, considerari potest, ut habeat
fe
solum unam para Metium, Teliqui a, reliquo enim literae
c conciderari possunt ut a, imultiplicat per cerlum numerum B
 certum numerum, ut  b = Ba, et c = Va, 
Hino  ex omni aequatione ad curvam absoluta.
c ordinaria, duci at differentialis carens parametro, seu
in qua la X et y servant vel solae, legem homogeneorum.
quanquam ea saepissime erit irrationalis, et redcenda ad ipsa unde
ducta est per sigular quadraturae genus, modo explicato. Vicissi
Rusus omnis oqualio differentialis primi gradus ot reduci ad
differentio- differentialem, in qua solum X, Y, ax dus. dux ddue
Et qua in arbitrio est assumere rclationem inter Z, dze et dZ, ponendo
ulcunque dari ex X et y. si hac arte posss assumendo novam
salem vel nova aequalionem pro hac clave, possemus tollere
simul X et y haberemus rem reductam ad quadraturas relavem
inter et dy dx et dy,; eax interventu quadraturarum
sit X ddx + yddy + dx - du =0
unde demus et inter x et  qua habem
2
ex dex = ex x Lxtx: xdex + dX sel Lxdx + aydus - 2 dus= 0 fiet
faciamus dydy- 2 dus= dyydgia, quod in nostra potestate
est et ex 5y. 3 tiet dxdx + Lydy Jus dy + d yydy:a =
seu fiet xax. +yydy: a B =0. Et statim potuissel
t procedi in eg. 1. faciendo yddy- dyt = d dis dus
set
si placet. (servanda scilicet lex homogeneorum transcondentium)
abetur, et fiet Xdx  + ydy  = 0
a
quia Pxddx +sdu FxAx
seu si  dy y day - dys = dydys seu = y y ddy 4 dy
fiet 2 di =0 ergo foret y constans. contra intentum
sed si faciimus y ddu-dus= dyydus: a fiet yddus - dus = yydaus: a + 2ydes: a
seu fit dy = ya egy adque videndum quid tali casu sit
29+a
constans sit autem duus: du =
dy. Eyta: ya-gy. seu fil
log. ius = Quantituli, quae haberi pt ex log. y.
si dy = 6y de posita dE copstante sunt y progo geon. sed seu 1ddy edi
sed qualis haec proscessio? ubi dus = ya -yy adus sed de his jam
alibi dixcmus
cyra
axydux cexdy. =0 d et differentio iff.
(2)
erit a xx, ddus + 2axax dy + ce x, ddy + cc dxdy = 0
assumatur mxX 1) + n X g, cujus diff = m xr dy + a 2 m x ydx + nxay et 
et differff. = in Xxddus remX dxdus + 2m X dxdy t 2m Y (x rndnius e ralus
si de sit ostans = dx ... . + dys u. (1)
si proponatur AXXdy +cCxdus = 0. scribatur aXXdy + 24X ydX + Ce Xdu - 2a L
et summando fiet: axXy + cC xy = 2asxydx
Tantum ergo opus est sic
interpretari X et y ut sxydx sit summabilis. sit ergo X =x
H y = q  fiet Gxydx = e Zm n  sque + tiqu etm n v dv
petm pen aZe + h. etm v  quae quantitas ut sit
Summabilus videamus quo debeat esse relatio, nempe dZ v = 12 v dZ
+  Z v de posilo ergo e - b, et h -C, fiet e 1 et fit  et sit
L-1 7C ∞ ergo debel esse C+ h - 1 = D (+ n - 1 (F h + n) = (e+ m) + (2h ( B)1
sed ita bit h= 0 quod ab P= h ergo ergo m =0, etc. Erqua
res non procedit.
a = y CG X log. a = log. y = Say: Y ergo dx= dy: y
et d y = ax 7ga et dy - aa = a. lx. log. a