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# Title: Leibniz-Handschriften zur Mathematik LH 35, 6, 6
# Folio 37r, page id 00068051:0073
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vel
C
C2
In XXdx + ya dy = Xydx sumucando fit 2x +  agy = sxydx + a
Aequ. 1. Parsus dividendo per y prius fiet XXdx: V + ady = X dx et summando st
fit: Ixxax:y + ag = LXX+ naa _ conjungendo duas ex aequationibus
tollamus a, nam ex equ. 1. est a = Xydx - XXdX : ydy. et aq = XXyyF -
27X 2X9dx + XaX 3: gud quibus substitutis in aequ. 4 fiet
XXAx: y F Xy dxs XXydxdy L L XX yy dy + 1 aagu dy uxxyy ax -2. 1 y d  1a EX
y dus
Hanc aequademo diferentiando (prius divisam per yydus differentiando sublatis
summius, habebitur differentio. differendiali(8) in qua y, X, du, dx, daus dax.
ut ergo ope aequ. 1 et 4, habetur aequ. 8 differentio differentralis, ita ex aequ. 1 et 2
habeti potest alia def sublata prius a, haberi ut alia differentio differentialis s in qua
parsus serunt tantum y, X,  dy, dx, dau, ddy. Et quidem in unaerit quantitas
numeru 1, in altera (9) eut quanditas  numerus 10. non ergo apparet quomodo An ergo
C
viceraus 10 et 4 talem inter se habere debere dependentiam, ut coincidant agu 
∞ aegu. 8 et 7. Iten Et ita esse suspicor, nos tamen hoc dissimulata potest
tamen caliquid contradic ut scilicet eadem utrabeq sit linea. Tollamnsantem
sed hoc dissimulato si ope aeg9. 8 et g. tolleremus y et faceremus deinde
Xdax = dx., res ois esset reducta la  dx du dax, dagu. et problema
esset solulum per quadraturas. sed calculus reapse tentandus esset
bic duani partiam aequatione XXd = ay ax fiet
nam ob dictam ravem ei non fido.
facillimsa
dividendo per X fet dy = ydx: XX et y 4 10 a a Sydx: Xx Rursus aequ. 1.
(3
tentemm in
Ex 1 et 3 Collendo a sit  y + 10xXdy: y dx = XXdy: ydx syax: xx seu
Vis dx .= 10XXdy = XX dy sydx: X, et differentiando fiet:
 seu yy dx + 10 = (ydX: XX, sed ita tollendo summam simul Collitur 10.
hiet X XXdy gy ddx + XXdy 2ydy dx - 7ydX XX ddyue = y dx 2X dx dy =y ax: XX
aotre
lo
e
rividendo per 4, f et na per)  fiet XXdy: y = adx unde sxau:= C aX + Ha
XX ay
(10)
seu X d Ny (Bdax +2X dus ax - Xeus ex daye - 2 X a) ax ay = (X dy ax
seu ex aeq. 1 differentiando ut tollatur a est yax XB dtqus 4 ydx 2X dxdys - X dys yd dx Aay =0
que dicae consentiunt quae duae eh9. consentiunt
a
valor ipsius a ex aeq1 in aeq. 5 fiet SXFag. y C X Ely: _ ex +11X tys: yd)
quam diffeuntiando siet X dy: y= ydx X ddus +ydx 3X dxdy = Xdyydax. X dy dy dx 
e
y di
+ X 11 s dx Xiddy + 11G dx 4 X. xdy - 11X dy 5 2dX dax - 1 X ly2 fay ax
L a
3x 
gX axdus = x dx ddys +3) X Ax Ay - X y dxdy dax - X yitys lx f 11X y cxdus ca . Xy ex  is3 5 x coida tus
e
seu
dgEgnde et