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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folio 25r, page id 00068196:0049
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Pro A differenti saetem pro parte, ad Terminos serici rediri
potest. aadue = XXdx + gydr.  es)L aay sit
aa ddy = XX 2x dxdx + XX ddx + 2ydydx + yyddx est autem sel
_ dy dy =  () - y dedbus = 4y) (5), et fiet Tollatur dy fiet.
aa, (y) (7) = 2x dx x + xX dex + 2yd xf(y) y) + gyddx (∞) C
addus = 24 dx ux + aa xx ddx + 2dx: 2 yxx dx + 2y dxdx + aa yy ddx
dy = (y) - y 5 (dy) = D + ddy = (yD - (y) et ddy = (y) - (y) - (y) + y = (y) - 25y) + y
(dysidddu = dd (dy) = (y) (3) et dE. d’y = (Iy )) - (y) sI) e    ((,) + (y)
ddy = (dy) - dy = (y) - (y) - (y) + y = (y) - 2 (y) + y
dy = (dy  odys) = ddu = (dy)) - 2(dy) + (dy) - (y))) () - 2 (y)) + 2(7) + (y) - y
dy -. dy = (y) -y
seu d’y = (() 6 (Cy ((y)) - 3() + 3(3) -y
et d’y = (y)) - 4 ((y1) + 6(yy - 4 (y) + y. Et ita porroin infinitum.
similiter (7) = P+ dy (y)) = ddy + 2(y) = y ( = ddy + 2y + 2dy - y = ddy +2dy + y
et dy = et (Ry)) = d’y + 3 d dy + 2.3 dy + 3y = 3 dy = 3y + y = d3y + 3ddy + 3dy + y
CVy
et eadem modo () d’y + 4 d’y +6 ddy + 4 dy + y et ila porro.
Bdy +Vddy sic explicari debent ut tollatur () fiet β(y) - By+ ) (1is oces
(aapyse pay= a dyxa dx et Gya
_) sit  B = 27 et prudibit (dy +2 Bddus = 2D 2B((y)
Et quia ex problemate habentur dus, ddus, d’y etc. ex primis Xydy,
(yD), etc habentur solas
il
accedentibus var dx, ddx, etc hinc (9) (7),
si dy fuit prog si dy ddy d’y d’y essent progressionis
primas Y, et dus, et progressus ipsuis X
14
beomeliveae etiam Y, (9), (9), etc progressionis Geometricoe forent
A X doge = dE dZ deZ d d2d ddv + de
dvq) = d vd Z + avdeZ  dd (1Z) = d vdZ 2d vdZ + d ssa 2
e
Z d (VZ = d v dZ + 3 dvd Z + 3d D d Z+ d’vd 
e (uu) =  Z dvd
1.2
Hinc dXxx = de ExXdx) = PXX L L sex de Hxt 28 X d  + 2. C- 12 4
que ete
de _ axX) = 2Xdx c XX = 2X ddx t2dx dx  d’xx = 2 X de+ 3) deddx F 2d xdax
et generaliter d
I5 XX = 2  2 X dX + 2dx dX + 2dx d X + 2ddx d3x + 4 dx d Ex + 4 ddx d X + B dex dex
Xx =2X d X F2 dxd3x + 4dx d’X + 4 ddxdax
a
A d’Xp = 21 Xdx d’ XX = 2X ddx + _B N 2 dx d 2, Xddx + dudx
= 6 2. d Xdx
d3XX = 2, x dX + dx ddx + 2 dx ddx = 2, Xd3x + 3 dx dax = 2, d xdx
seu dsXX = 2X dX + lodX d
d+ xx = 2, _ XdEx + qdx d3 +3 dexdex d’xx = 2, nd  5 dxdx HC ddx d3X = 2, d’ Xax
X XX = 2, X d’X + 6dx d5X + 15 dexd X + 10 d’X dX = 2, d3, Xdx
N XX = 2, Xd EX + 7 dxdX + 21 dax de x + 35 dXd X  = 2, d, xdx
6
d’.Xdx = + X ddx + 1 dxdx
d, Xdx = + X deX + 3 dxddx
d, Xdx = + Xd X + 4 dX d3 f 3 ddxddx
a
de, Xdx = + X deX + Sdxd x +19 ddx dX
d, xdx = +X dbx + 6 dxdsx F15 cdx d EX F 10d XdX
d’yxdx = + X d X + 7 dxd X + 21 ddxdsX + 35 dex d X
5. 8. G
G. 6. 7 38
Ex his patet regula progressionis
Numeri scilicet cuiusque columnd
demto primo in columna
sunt ordine sumti naturales
Triangulares
d, Xdx = + X dBx + 8 dx dx +28 ddxde X + 56 ddX + 35 d X d
, xax = +X d ∞ +9 dx adx + 36 dlx 47 x + 84 d x dx + 70 d x d5
et
exex 
pystamidales e
d
M
etS
et  f, eA-f, ek
L. 2. 3 ete usque ad f
nisi cum e+1-f =F. _C
seu et 1=2F, tun
cur dii-
10-4.3
f0 2203
iu trais
4. 5. 8