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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 3 (shelfmark LH 13, 3)
# Folio 116v, page id 00068200:0234
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Verum brevior haec operatio: sit data Xb + yC + X. yh
quaesita
X e + y f + X. 9
Assumatur, aequatio: X L. + ym + X. yB aequ. 0. cujus diffecoentialis ent
liqu
Cujus differentialis erit: dl ax + dmdus + dn dx + (du) dy. aequ.0. seu fiet
x (4 dm +(dn). aequetur  hic valor valori aequationis, et multiplicatus
B aequ
dus
a1 +) du
terminis er fractionum per crucem, fiet:
X b d 1, 4 yc dm (+ xe dm + yc dl + X b du tyc du + Xghant refentes sela
X b +1) + Xedm + Xb du + Xe(dn) aequ. 0.
yC dl. + yf du + yc dn + yf (dn)
xgh dl + X ygam X. yh dn + X. yB (d 1)
I qua aequatione sta omnes termini, qui componunt xb dl habent solam literam X
non vero literam, y. Et omnes termini qui componunt yf dm habent solam
literam y non vero literam X. Reliquae quantitates  omnes aequationis 5
componuntur ex  terminis habentibus tam y quam X. rebi tamen ad huc
Superest aliquid annotantum X. yn poni potest aequ. Xy X p + xy Y 9+  XG2n Cujus differentalisent, D egu a sospitdus hyt ry B1 Exry  gi ex) dscx A rus dgudus ex Bstx tudidus
annotandum aliquid Circa reliquas aequationis 5quantitates
Ergo d X. yn seu dn d x + (da)dus erit aequ. y xp dx F X xpdy + y y gdx + Xypot, tyXy dindr  +x. y
60
seu du aqu y xp + yy9. et (d n) aequ. X x p + Xyq
et similiter potest X. y in poni aequ. X XIp + Xyy2q + X9 X. y3n
Xy
et eodem modo X. 93n _ aqu xy X3P +Xy y39 + Xy F y 4n
ubi tamen aliquando si aequatio 2 sit finita, finienda erit ipsorum Xyn
progressio. Hinc fiet
X. yn aequ Xy XP +xy99 +XXyy XEp + Xa ry y29 + X Xy X3p + Xy y39. 8C
memorum
Ubi tamen fieri potest; ut aliquo terminus, ut Xxgypq sit aequ. 0.
c107
Htuc jam d X. yn, seu dn dx + (dn) cu erit aequ.
=aequ. Xy. dp dx + X. Xp dy +GXp ax + Xy. d9 dy + X. y9 dy F y. y9 dx
g
Xrgy dip dx + pd + EX VyxpB L Xxgu l Adus + Xxy iq
2Xky Lepas
Xy dep dx +2X y. Xip y + 2X1 XEp dx Xy. d29dy + 2X y. yequy +2xy y2qdx
XY d3p Ax +3Xy. X3pdy + 2X. XBpdx) Xy. d 39 dy +3X y. y39dy +3xJ. y79 dx
et
nunc quidem nondum necessaria est haec distinctior explicatio exta sed sufficit aeqat
Xy agdy + X. y9 dy + y. 79 ax
et fiet an aequ xy dp t yxp+ din
aqu d xy Xy 2n
et (d n) aequ X et fiet du aequ. y. yg + y. xp + xy dp + dEn
d2n r 22)
qui duo valores ex aequationibus 12 et 13. substitui debent inaequatione S
foret aequ. I X. 923)
(du) aequ. X. Xp + X. y9 + Xy dq + (d 2n)
aequatio 6ta, et erit  d X. yn es seu dn ax + (du) dus erit aequ:  xio dBiry +ysodgiry + xaodgrry
) X y dp dx + X. X p dy + y. xpax +  dEn ax + (d  n) du
ea
Sed quoniam ex omnibus his terminis duo tantum excerpi possunt, X. XP aus et yyq 
ubi sola est linerax vel sola litera y, sublatis scilicet dy et dx. hinc  nostro scopo
C11)