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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folio 371v, page id 00068196:0738
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An ergo aliter tentemus verssumendo primam aequo pagiuae scedentis:
aa du +x ydy + xydx + aydx = axdx quaeratur X per V, et sit
XC 10+ Wy + 1y+ 1393+14 y4 15y+ 16y ex et X = 10 + 210119  2ronus et fre 
Xydy = 10Y + 11y + 12 y3 + 13 y + 14 15 +15 y
aydx =a, 1.1, +2.12 +3.13 +414 +5.15 4616
xdxte pion ru asuiva et. eu ert
Eiri
Bxip -i 12. 10u1 25 0  3( 2 . 1
aa = - aa
20.
si fiat Zydu taydx-axdx +yxdx radys-
opt; s qe qua differet d valore
ipsius X nisi quod 10-0, videamus an
aliqud hinc duci possit
Nempe si eq aequ. 4. suferas + 10y
=a, 1, 2. 10. 11 2. 2. 18. 12 y P3. 2. 10. 13)14. 2. 10. 1415 ey
XY
. 2. 101ry +2. 2 1c. 12( 4 3. 2. 10. 13 1 Cx
seu 410)  =a, 2. 10 aX + L. 2.10 ydx.
residuu inaeq. n 4 non differet ab aeq. resullante
ex explicatione aequationis 5
sed saccurit alium scilicel 5g. 1. et 2 habent eandem diferentiu differentialem; per
quam invenietur, a quatio communis utrique et valor etiam ipsius X Mer y ubri
fiet
 quatio ota
et X   20 +21y 4 22  + 23 y et.
 primariarm albesariar
, 2 21 4, 22y + 3. 233 +9. 24y  tC ixdx data Ny dyt agdx axdx tyx  -aquo
 104 11y. + 12 y etc
dp cndy + 2L vd 5 13y a dy estc
+ Xyay = ( X 10y + 11y + 12 y + 13y714y+15 y etc
etc
fdx
L.HI. 2.12. 3.13 4. 14, 5.1 6.16
= aXdx = -a, 1. 21 +2.1 43. 23 1914. 45.25 6. 26 7. 27
+ aydx =a,
etc
dlax = 1. 2 diy + 2.3ydy 13. 4ydis et
quatuor
+ aa = taa
+ yixdu-
121 222 323 4. 24 5.25 6. 26 ex
+ y dxx 0
a dx dya.
Fagdax =a,
=ar  1
Unde talis oritur aequatio
10 tuy tres r3y ay +15y 116Y etc
3.13. 4.13 3.18 si 6.19
121 23.13 3.4.14 4515 5.6.16 6.7. 17 etc
1n hE
Hinc jam per aequationes compa
+22 2323, 3.414 4525 5(. 2b 6.7. 27 7. 828 eH.
1.21, 22 3.23 414 5.25 6. 26 7. 27 eu
ratitias O destructitias
sunt tolli
O. D, 3 et G invemeri possu
tres literae 22, 13, 24 bm
et habetur relatio inter
Notui autem
11, 12, 13
10 et 21 ingredi calculum,
ut scilicet servatur generalitos
12 2 2315 3414 4515 5616 67 27 ete
Txdxdy = h
ut vero hic calculus sit communis omnib  ribus literis impediatis (demta
prima 10) sit 2= n siet 3- n+1 4 = n + 2 1 = n - 1, 1, 2 = nn -n 23 = n + R 3.4. n13, +2
cmttishi term
+ylx = 2
Et ita putcherrime et brevissime habetur regula generalis fimita pro terminoru e
potuissernus etiam aequationem multialicate
per vel per ay vel per simile aliquid, pro arbitur, unde rursus novae aequationes. Et videndu
an sic  obtneri possitet tales aequationes ut composititae quasi ntio aequatio esset ordinarium
Quin et certo casu poterunt
aet
progressione, non alioqui facile epuenda
Javenori quadraburae speciates, ~ o. Esto aequatio axdus + aa axdus + baxdy
explicata, ut si semper simul exearent a1 + b21
ax dy + baxdus + caedus + Cy dx B.5 a dx =  ft Ergo in casu quo es -5 = 0. sit
iterum Ax + Iax + ca = 0 quod quidem ex ipsa corva patet eu differentiali: sed securius habet.
ex seriebus. Et ita invenitur tam y quam X. sit a Saa + XX a et + est dx: fdX
E 2Xax = Cy Ax - fdx + Saa + xx dy = 0 Ergo quo casu ey - f -
tiet y Fa 2Xdx: taa + XX = dus sed an licet hinc inferre? esse aatX = y + b
eo iesa quo ey - f 20
videndum quo modo ex seriebus talia jastificentur, a quatenus
x = 1. 2. atus r2. 31. 19 iy 13 424y dy = (p 6) 2. (X ax
Fairdys
quae ratur differentio differentialis aequationis B1 fiet: xdus Fi dedtus + a gadx = a d pdx A Xte ay
+ yxdx = 2)
+ Xdu =
et