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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 2c
# Folio 12v, page id 00068199:0024
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ay: dx = xx Fgy: aa  seu gy: aa = dus : dx + xx: aa et y =  Ysaa dydx + Xx da dx)
Si ly=
a dxdx
...) +
aa dyadx + _xa LX ddue + 2X dx + 2 xx dx lx
2adx YC,...)
Aax dax (uas X FXptnlx)) (Taa yrelutl -. aa dxddus lx - 2X  X dlx ( 2. X . t dx llx)
alte
qu aa Aysd dex t aa (lx dx ldys + 2x d P + 2 X Xdatx) ddx
aadysdx xxdxdx
a dy dxd ddydax.
4qu (aatdy ddx + aa dxdx ditus + 2 dx + 2XX dx dx dd x)
4, aadydx + Xx dxdx
aady = Xx dx + yydx fiot aatus aa dxdius - a adyddx exdxtxrax
dxdx
2Xdx F2ydy
aa d X ddys = aa dy ddx = 2x dx - y = aa dy xX du lx)
2 dxdx dys
a xx dlus dibus 2 2 a dxdys dux dius - + aa x dud ddus + a du dydliexdd - 4 ca X  X du Llx + 4 XX d e
Paequ. Aaa dxdus = 4Xx dxd dydus 1. Compendio
L12r dy3
100. + 11I dy + 2 doly +dP dy.ly F, 12 cldy + 122d dy ddy = 0
ubi 100 = 4 XX d x6 et 11I = = +aa xd  ddx et 121 = at ddx lax + 4 XXdX et 131 - - q aace
et 100 + 110 du + 120 dydy + 130dy + 10) ddy + , 11 dy adys + 102 dy ddy = 0
fiet 100 = a xx dx et 110 = ( 4aa X dx dd X et 120 = (a dox dx, 4 XX dx ut 130 = - +aadX
et 101  = = 4 aa X dx et 111 = - 2a d X ddx = 0 et 102 = a d xd X ubi ponanus dix =0
fiat ly = 4Z. Z et ddy = Z dazue - d7d z , : Z2 ddy ddy = 22 dd zal aze = 27 447 laZ + 424,: 24
=  - d7, Z -1 et dy ddy = Z 4Zd 1Z - 423: 23: 1. Et fiat jam
et
Z d (dZ. Z) sunuuri.
peest
2
=Csvix dz = Bdx ( sax 1 daque - duax b   +rlxd x
ind potius fit  quod eodem redit pial initio tati-
du vdx du = vdx, et habetur aequatio d
unu 3 differentialis uno gradu altior posita ax est aute. si abest
hest
ba
 in aeq. differentio differentiali sunt quantitates dy: dx et als. adx
X ddx: dxdx , et X: a fiat X. a = ZV et dy: dx = AV: Z
et X: a = Sdz: Z et _ et tZ d x = asdZ: Z et dx = adz: Z et ddx = a d (2)
et Xddx: dxdx = A(Z: Z) d (az. Z): (d7 dq: 22) et ddus  =  afi ad 2 d (X) + a 1 d2) ~
e ddus: daZ = (dZ. 2) d v. Z) + d dZ. Z) v: Z 3 :, d (dZ. 2)
ita tollitur a
22
porro non habetur ddv. Ergo ponendo ddZ = 0, habetur 59. E differentialis ubi
est x Z: V, de: d et (dZ. g
idem est initro in differenticali ot dinania fiat X: a = fdz: Z, et y: a = v: Z
habesitur aequatio in qua erunt v:Z, SaZ. Z et de: 4Z
 2  et.  = b et v -  cobt fit ftz.Z) = t
su vi co: t sit Z- ad dZ = 4 d Z dco: a
et dys dy = L
aute
seu dydys
scop
dgrx dy: dx
ded
dlsa h 2do2t R
d2
daqu neque
1
M. 2
-adx