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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folio 374r, page id 00068196:0743
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+= y = dx: dy = aa +Xy,:, ax-xy -ay seu y aady + xydy + xydx + ay dx - axtp -0
Si XC 10 + nyFny + B9 + 14y + 15y+ 16 y
X~ 20+219 + 2yF239 tury +29+2696
dx = 1 Ut 2. 12y +3. 1394 4.149 4+ 5. 15y 4 6.16 5 4, 717y
ax (1. 2 + 2. 22y +3. 237 +9. 2419 + 3. 25yt +6. 261 +7. 27 y xap
dex 121 25, 1 3+. 147 ddx = 1. 2. 12 + 2. 3. 13  3.4. 1y + 4. 5. 15e etc.  et denique
ddx = 1. Et dax = 1. 2. 22 + 2. 3. 237 +3,4. 245 + 4, 5. 25 is etc = 2 dxdx
+ Xydy = X 10y + 111 + 12 1 +13y+ 14 y +15Y
+4y dx - = a, X 111 212 3.13 414 sit5 6.16
=aX dx =a, 1.21 2.22 3.23 4.24 5.25 6.26 7.27)
+X9dx = 2
1.21 2.32 323 4.24 5. 25 6. 26
+aady- aa
axdytus
xd cenri
(10
Quaeratur sam differentialis ipsius1
=0 fiet:
O Xdlys + ijdxdus L ndi dyt yd xtx taijuprajib, atu
vel saltem ex valoribus 1q. 10. a dyde
differentutos O=
o= d xyilus
Xotis = Wo any y  + 16V dxydy = 1 My +BEy a +5141 b2s
ydadus 11 in 2  +B 41t 15 616 dayix = a, 1.1 4. 12 913 (16)14 25. 15 3616 62)
atrs a dxau = a, 1u X12 3B3 4. 14 515 626 f. 17) -adxap = - a 12. t 23 23 24. 24 4815 56266Y. it
Tayddx = a, 17(2 2313 3714 4,515, 5. b.16. 6.7. 175
adyex 7 Ra 412 913 1614 25.1 36. 16 49 17 a xyix = 1, 1. 4 4. 2 923 1624 Cet. 15 36. 26)
b
Ex aequationibus 10 et 12 excerpartur aequationes destructitiae o, . quarum ope tolli
poterunt quatuor tres literae, 22, 23,24. Ut autem  incipio autem abtus, quia sic non
ings inpeditur 10 in quo est varietas. Nam post O vel S in infinitum omnia euut
eoc tondo C a  Ut autem calculus sit generalis, facciemus pro numeris veris
2
14, 1.2 n 2. 313.4
Coepficientibus 2, Z, A, substi tueraga., ) n+ 1( nt 2 nnan unem nn+3n+2. Et fier
D Eitnan  = a n+ 12Z + nEL= D42I + 2II113 -n+  + nF 123 L0
id jam video 3 et 4 nihil prodesse cum nihil aliuo sint quam ipsae C et 3
8 2u2 duplicatae.
si distructius scribatur
itaque redeamus ad ae9. 11. quae erit talis
=a d xax
Xdus + ydxdy + ay ddx + adxdus = axia antx + X Ydix Xapayr y axdu 2 0
a
Mattiplicetuur aequ. 1. per , quia et ista dis fiet ex aqu. 1.
ac aagydus tadg adu t xydgydus + pydus adque +xydrydque + xydxady + aydx ydus taydxadus
atque yy aedy
lccet
-axdx ydus axdyadus L0 jam pro gaxdu
cde
posum,s quidem aqua quia autem haec aequatio differentio. differentiatis
non hac aequationes nisc jam lalentes in differentiali; ideo cogitaevi posse
novas obtrueri si differentaetur esqu. 1.  ut differentiando tollatur X
Nim ex aequ. 1. est X =
nady+aydx: adx y ay ydx unde
dx a dx dx = 2 aydydxdx - 2ay dx dx + y dydx f 2ist dus de + yydx dy
un
Caydux ddx= ayddxydy = aydax ydx) - ax addx aadus = addx aydx
sed ibi assangitur ad terminos uubi se valores ipsius X
valoris coefficientes, hi enim lalent in dxh. Quod si aequ. 1 multipliceuus pert
= y ddx aady (+ yddx aydx
et productum differentiemus vel ex ag. 11 tale productum fabticemus. pote rit mjus
ope giuu hanc ipsam differentiationem fatricemus; et deinde hanc qulingamus
cum t. 1s videtur tolli posse dxh sed relect ne in novim aliquod tale nos induamus
X = 204 21y + 22y + 23 1s +24y eu
da sit ipsius x defficiens vocetur a 10 ipsius X vocetur 2OG et X vocelur 3C
X X = 10 + 1y + c2 ist et
d Naneus designans pr gradum vocetur seu exponentem 
coefficient iatiu
dx = fn 11 nV
ipsi Numerus gradus seu exponens e sit e
sic ax = 3X AX = n31 2-1
dX =2Xdx = n. 21 
X = 10. 10 + 2. 10 1y + 2. 10. (2 est  + 2. 10. 13 1 + 210.4 y aet
11.
dx = u u + au L. 12y+2. 13. I.B3y= +  etc Ergo dx = ( sed hine vndeo non posse dx exprimiper
Ax = e H a M
2d X dx =  X = 2X dlx +2aX
2. 2. 2. 12. 12..
respondentibus terminis
etsi eare e
veri coeffirientes ubique bariant
ingrediente se, nume
11. 13 Ct
ut in yEs
in hii st, haebel 11. 13 habet 1.3, et 12. 1.
habet 27. Et i   institei derivatio in summa,
sius ax
itaque video non posse hac methodo magni aliquod ngi