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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folio 369r, page id 00068196:0733
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1)
Ly: dx = a. Xy + X. y seu dy: dx = a i +  n. fiat seu dy = . M dx + n dy
e
fiat M dx = X deo e et sit du constans, fiet du = a X deo +ndu
per 6
exatt m dx. X = deo seu d m: dx. X =0. rurrus ottys-nix: do a
3)
et fiet d u ndx d X dy-ndx 2d  0 ped nihil sed hinc video
non posse X et y. simul tolli, nisi ad hibita aci ex aeg. 7 Janc si faciamus
praetetea a = du - ndx:  x et disferentiomus, nihil prodibat
novi 1 1
illud video, si sublata una litera verb. gr. X
in aequ. differentio differentiali nihilominus alia, at a maneret
foret unius gradus, tunc hac divulsa in duns evanescere posse
sed dubrum an hoc liceadi
(1t
q
EF. posito f dari ex x, 9, a. differentietur
5g. 2. ut tollatur a. talum equatio lni qua f, df, x, dr y, dy Hanc differenti
opeaequs 1 et 3 lollatur f fiet aeg, (4) in qua ds, x, dx, y, dy. Hanc
Herentiando tollatur By et fiet ut aeq. in qua dfd, X, dx sdx, dus, adus
sed nihil hinc
a dy. +x = C_ B xet F 1oR + 1X + 2y4 HXg + 1X + 22y, :, 3 + 31X + ry eu.
12
10 Xy dx +, X ax = 1 X dy 12ay dy sit rydx constans = a deo
fiet aco= Xe2ay  - 2 11. XF. cursus Xydex + 2Xydx + xay dx = 0
denique - 11 Xddx 3. 11X x 4 12a y Idy +. 2. 12 aydy 20 Nihil
10 X dx + n V dy 4 a X dy fit: 10x ddx +910X x 4 11 1 ddy + 3. 11 y dus = a Xddy+ a dxdus
a = ul
b - Utm
10X dus ddx + (3, 10 X dx dus ( 11 1 X dyaldy. .+ 3. 11y X dis = 10X deddus (110 X . dyse in) aerctus
n
n
mAn A p
c~ l FEmtM np
mtnto
pra
d = liymtsnip nepe
intonBpq
e = Uant burrest
+uy detus
Methodus haec videtur adhibenda: inventa reguta
progessim supdatur d ex 2y, a. inveni
pountur dy. ex dx, X, y, a. ponatur dlx = a. unde
et evanescent differe ptiae reliquae, et quae ratur ddque
dadus etc. ex iisdem postremoa inveniaterlex
a a- l aubam abte n ansbesenduq
b= 1m c= lem  +m +n seu ea lt im+n et
d lemintp. Et ita porro
1 sit aequatio (1) inter HiX, _, dx, dy:
i aequo. (2) inter ax adp ddy.
aeq. (3) inter y, X d X, ddx dadus
aea (4.) inter y, X, ax, dax dddx, ddddx, dalus
eteere
ien dantur per 1 valor dy
in
per E valor daque
per 3 valor, dady
etc
Hinc psy, day, ddaus esc sumendo pro m, n, Bet
et
, (4) (y) et sumendo pro a, b, c et
poferimus per regulam scriplam ipsas (y), Ay1) etc.
formare per ipsas dus, day, erc leu per earum
valores, aB est per y, X,  dx dix, erc. accedente.
confrante a forsan constante a.
Hinc ssi y X primas assumamus abdete non ut
quaslibet seu constantes, seu in dataratione ad a
et assumamus etiam certam velationem inter
Xde, dax vel solas vel cum interventu constantis
vitendum est an simplex quaedam velatio inter
progediendi in inde horum ope inveniantur
(9), ((7)) ((7.)) etc done inveniatur lex propedi
y, ex primis X, y a. priaque ita primae X, y, aity (9) et (yD haberi posset progressio obtineni
posit valorum ipsarum Runilo   ex ak
et de. seu inter a, et
endi. ita habelitur modus habendi infinitesimae
revera eruat constantes et prout assumuntur defer
minabitur punctum per quod linea transire debet
Raper ita reperitur valor generalis cujuscunque y
parvam cotantem irccedentibus solum numeris. Quil
et iasi ceu certo modo revecari potcun
ad (deo, seu ad co, seu ad numerum
ubi reperta generali relatione, haberetar aequad
ex constantibus assumta distantia, a prima, quae
veterminandi ordinatam. Haec lex si non est reducibilus, altem transcendent. ipsi numeri coefficientes habens
tanscet
in valoribus  b, C e. b, C, d, e (tollegolas
latet simper in eoefficientibus. Et ita habebitur novus modi
ad aequationem ne transcendentam quidem noenipatur ex
ne es as de abrit patatur inter de combii atun eeta  s
senu
camnova
n  a aad bl s et  inter se co
a
bac
e
a-2b+c=n
a -3b +3C0d
b. utd b. 3c +id. C
c- 1d+e
4b +6c - 4d + C29
c
cad