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# Title: Leibniz-Handschriften zur Mathematik LH 35, 5, 25
# Folio 13r, page id 00068043:0025
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Quomoa differentialis implicata reducatur ad absolutam seu quadratoriam, quando valor Ax: dy exa per aliquod ut X. Y solum
BA (AX: dy = i t. pona autem  posse haberi ex solo  X: y
cetterminar ponst.
 exee seu eC= set seu posito X: y = Y esse + - Y
digen)
prose.
dico posse aequationem certulea differentiatem reduci ad quadratoriam
cutiones
sit enim Est enim dY_ seu d X: y = +y dx + x dy,: yy A
s
1
= dy + dx: dy + X: y e3: y =7
E = Y,  dy: y. Ergo dr: + + +Y = dy: y
est autem + = B Y. Ergo sit Ar: + Or + Y = dy: y. adeoque
datur relatio inter Y et, Y, adeoque et inter x et y. per quadraturas
ponamus dari + per x y sit qu xy = m. jam di
Rursus. dx: ly = + + = e) m  posito m =X y.
ei
e
am = X du + ydx s4bG Y dus xiy + duou dm Y
(1)
seu dm = dy Xy + yy dx: dy: y. = ds m  dy m: y + yt. seu em= dyn: y+ dyyt
2
tursus dn = dy o m: X + pt dm= dx m: y + xt seu am= ex + dx xt ex mxpo
loco aequationis serbatur am = dym: y +du m: x F
4
et loco aequationis escribatur am = dx mx + dx m: y +
videamus an ope harum duam 3 et 4 vel 1 et 2 possimus
tollere X et y ex aeq. am = Xdy+ y dx Exoy. 1 est g- dm- v.7
dy
de
+dy
Ex aequ. 1. sit: x
dm:tayue M.y. dm: tdx = m: X
adeoque sit dm
m
X + dx + dx m =   X dm
seu X - L dm
2
dx. F
VEA am
ar
seu XC - dm X + am = L am - d m
 m
4dx f2
s
+dx
b
seu x= dm 4iam 14tm dx
 2F4x
etqus = et similiter y =d m + idu a rem et ut sola supersit dus,
L
in aeg.  pro du scrbatur t dy. f Ergo ex aeq. dm = x dus + y ex fiet
2Edy
fiet per O et Y siet am= am +Y diu + 4 ttu  amq ram 2intutus
e
seu m 2F dm -
e
+
2t
2 + dy
 dm + +am + da + am F du +F sam +amtdus
non posse esse iidenticam ex eo patet
quod non destruitur dus, quando duae irrationales (per infinitam.
l
uam aequationem
extrahi fingerentur. Ergo sublatis irrationacibus habetur du in
denqe per d per ay: dm per + et M, quae ambo ex M haberi suppono
itaque cres reducta est ad quadratoriam
I
idendu