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# Title: Leibniz-Handschriften zur Mathematik LH 35, 6, 27
# Folio 7r, page id 00068072:0013
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generalius possemus assumere y b = y. c, ita ut c sit parameter in curva data
Nem nova. Nempe sit aequalio data ad curvam quaesitam dudaa dx= dy Xx dyt gydlus
et fiat X= b setiub = c. y et aa d= bbdy + gydy. posito ( Z = x f X - e seoto) dZ = dx
C
et quadrando fit aa Z + aaf =  sbbdus +  gy 5y sedd) X- e + f sbbdus +  9 ec
seu ponendo (guo) f - e = g fi x +9 = Sbbdus L L 73
Hinc si quaeramus jam concursum
duarum curvarum erunt quidem constans, sed x erit quidem 9 et c gemina se X et y unica
et fiet du = (mo)= 2babdu quod oidiamus an procedit in casibus particularibus, ut simbay, fie t
sudu  si b =c fit Sbbdu = sccdes = ccy et disferentiando secundum c, erit ascedus= cydc
sed E db bdus posito be C est etiam rCy dc. si bb = cc - gy fit fbb dus = scedlus - syydus = ccy - 1 y
et dfcetus asgydus) = reydus ,ut acte sit b- Cy+  CC3  fit fbbdus = cogys+ e + rey:aq
et cujus differentia secundum C est rcyyde + 40dc + 6ccyde, aa sed obd quod videam
an consentiat cum 2 db fbdus, seu afedus: a) feydus + sec dus :a = 2 (dy. a)
seu 2(yde + rede: a) (f cydys +sec dus 1a) seu 2(yde + ocde, a) (3 Cgy + cey, 1a)=
b  d fbbdu = 2sbabdus
22ye
Hinc si b = cy+ ce: a fit b0 dy = ccgytc + escydya
et bbdy:  3 CCy + ctus + ACyy3:aa, et hoc differentiando secundum X, sit
2 Cyde = 4Cydc + Bcc de gy, daa. Videamus quoan et quomodo consentiat cum
ab
2sbabdus. nempe explicando b, fit db (sec. C)  ydc +2cdc,: a, quae multiplicata in bdus
dat de, cyydy + ccydy + 2ccydy +2Cdy et fit 2s b abdus seu potius des b d dys =
2dc, Ecy + 3 cegy + 25y = de, 2 cy + 4 ey + 3ccgy ut ante. Itaque regu
reassumendo sic procedere opvrtet
sit, quaesita curva cujus aequatio differentialis data
Cimo
aadx = xxdy + yydy. fiat
(2d0) X = b et (3tio) b = C. y et pro curva nova ordinatim varianda
posito (5to) Z = X - e unde (6to) de = dx, quadrando
fiet aaque + aaf= (bb dy t, H  quam  - = (8V0) aax - e + f _ Eax +9
5bbdu +2y et differentiando secundum
jydus
(4h)
aa dz= bbdu +
mosiss
om
C et G, t manentibus x et y, ut habeatur concursus curvarum fit aadus = de feb ab c) dy
unde posis)f - C = g fit d aax + aaque=
cinmo
aa du -dc  = 2 de (b de c) b dy et ex aequ. 10. fol per 2 fit aabraaqus = (bbdly + Lyt
tiarum duarum aequationum ope tollenda est 2 y. t es quod ut ut E et habebitu.
aequatio in qua  b, db, g, du, et a, si possemus ita explicare  per 3 et C
et 9 per e et  per c, ut in proveniente aequatione tollatur aliqua ex his tuibus, b, g, a
dum tollitury. sed si nulla fiat explicatio. Ex sed hac dita
vlata autem explicato ipsorum  et b, ex 22 fint  aaqu=
a a ddg: de = 2 b dy d soc C)b) et aa db + aa dg= bodus + yydy
Cper y et lo
adque: de 2b d ((c. 1)6) Quia nil refert b explicemus per C et y, an vero  per
b + d9
a bb +7y
possemus quidem
b et C, ide fiat res hoc modo videtur perduci ad quadraturas
vel y per b, C sed polus opus est explicari b, sit sit b= y t suandet qua
a d((see 9b) = G fit aal ex 11, aodus - 2 de fy  du = 2 de 9,:y ret 1 = aadus
et ix va 1 fit   aag = (s ye dy + 2 9) ha y en  2y = au y
dratum
n12
10. sab  abste
=y concursus difecentiando idum b o et 2 sit a dbs +
ed
desecic) seddus
posito co = C y
est
de s cue eo