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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 2b
# Folio 16r, page id 00068198:0031
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Si detur dya = X dx . t sit ds ambigua erit y etiam ambigua
ejusdem gradus.. Nam quie libet ambignitatis ss casus habet suam propriam
integralem, ergo integralis hrabet totidem casus. Idem est non tantum
in du, sed etiam in ddys, vel in d y in genere, ut quot habet ambi-
guitates tot etiam y habeat ambognitates.
Sed quid si in valorem ipsius dus ingredititur y. Dico si g
vit Dico si y ingreditur libera ab ambignitate valorem ipsius du ingre-
aeuvation
ditur, tunc Dico si y ingreditur valorem ipsius du rationalem u
et y non est rationalis, sequitur  formatum ex y ingrediens in
Valorem ipsius du, non esse rationale; alioqui dus foret rationalis
Et rursus si formatum ex y est rationale, etiam y esse rationale
Et generiliter, si y ingreditur valorem  ipsius dus rationalem, sequitur
 prout y est rationale vel irrationale etiam ex ipso formatum
ingrediens valorem esse rationale vel irratinale
veabusr verli guotia ds dy = X ey gy) + X, Dico tiay-gu
st quantitos rationalis per X, etiam y tatum fore, nam si ay yus est
cntonalis per X, etiam du est rationalis per X, ex per aequabionera
hago etia in y est rationalis per X, ob regulam generatem
si psolationalis fdgiy vaque est rationalis nec irrationalis per X.
XY
Si  si sit aequatio 10 + yA = 0 sit differentando
dio + dag + _ dus sit substituatuo valor ipsius y= . 1
fiet dio  dutro + ndu fiet d10 + du =0
si sit aequatio 10 + 1y + u fiet duo + Duy + 2ydy + 11 dy =0
et dy =  - d10 - duy 3: 2y + 11 feree vel fiet
V = = - 11 dy = dio: det +2du, adeoque
10  10 -
11. 11dy + 11d10 + 1. 11 dydy + 2 deodudy + dio dio
du + 2dy
etfiet 10 du +2 1otus
=du du
dudr + et duc dy + ex dydy
12. 12 yy +11. 12 ay +10. 12aa
a 12. 129y + 11. 12 ay + 21a4 - 11 a - 10. 124
y = a FCu-4. 1012) = 13 + 4
10da + 11 dy + 12yy =0 sit 12 constans
16 diau + 4. 10 dusdys + 4. 10 dydus ( + H. 11
xcloulo
=edyrydus) fit dio aa tdu ay + + IIa dy+ 2. 12ydy, = 0
dio aa L 1 a du  a, 1b = 4. 10.12) - 11
E. 12
seu 2410. 12 a (F 2. 11. 12_ay) = 11 d 11a(+ 2. 1 +2 dy - d11 L(... ) a = 2. 12dyil..)
+2diodrdus fit y =
a
2 dy 11 d90
+ 11 dydus
du a + 2. 12 dy
Eudio dua fiat  2. 1
fial 2. 12 = 11 d1t = 2. 12 du seu 12 = LH d18,2 : xe, 90+ dio
fiet denique dy =
2
00
Et fiat du
2. 12 Y(152 4. 10. 12)
= d1
00 +. 1d11) = 1. Ergo si 12- 4.1. 12 vocetur C
le+ C SC W 00 F 11 dio -  10. 11 dir)
ae
ar
dC)
- du = dy Ergo
du = d dio aa +dn ay = iL 9 sav (na n’av +4. 1aa  + 4. 1r yy av = dio dio a + 2dueu ay t dnduay
4. 12 + (112 - 4. 10. 12
seu 
2. 12 1(1 - 4. 10. 12
nihil hinc utile ducitur
1a + 2. 129
fiet a1sc
p +2 dudu+ modv a -  av
susu arta dudir aa - At aV
=1flud1e