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# Title: Leibniz-Handschriften zur Mathematik LH 35, 12, 2
# Folio 180v, page id 00068195:0360
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Reneamus jam ad aequationem
gy - 2c9 + XX - 29X = + ff.
YX. AX X 91X XZYX  AZ- X, y
VG: YC = LGy. + ZZ +XX - 1ZX): 9
XX XZ = Z-X VYV. sit
V +g v = , Ly + ZZ+XX - 22∞)
vD = gy fZ7+ XX - 27X
y: Z- x = +: 
fiet Z- x = by. t 4 vn = gy +: lt) jy
fit v: y = (+ H). + ses fiet
y = tv: sC+H)  t (Z-X).  et fiat
In: (+H) e =X. Eg Ex aequatione
ad ciculum fit vr- ZZ + 22X  2c9-29X = +f
EF Ft. vocetur 44, fiat axv quo 2c v. 2992+ Egrv +ffq
JueC422+ 453 - 2c, tv + 2922- 2tVZ - 2992 egr = 0
+291V
seu Wo +ZZ - 2tVZ, 291 - 2Ct .x - = gZ  = + f
artitrariis aiginairiae f, I til, C, d, ftit artitiara per Y
adhuc artitrariam. Itaque quaevis fomuta in aequatoo localis hujus grodus fonnae
in qua WV et ZZ habent eundem coefficientem, huic identifiam potest
directirem
supra sectionem conicam retulimus ad rectam, quamcurque
per ordinatas tamen normales, et aequationem indenatam
 localam exposuimus. Possemus jam matare, eam matando
angulum in obliquum mmente hirectriae, sed pri
 nunc aut nunc at relineamus axem et am pro
directriae V. mutemus angulum. Aequatio oealis
Conica generalis id Axem ex vertice, est
 KR 10X. + (b: C) XX = yy. pro y
ponatur _ antc Er:g, ut pro x ponatur
2bZ 2bZ - at. g v 2 b1l. a B + (b: C) 2Z- 2 (b: C) (l: 9) 3n L u (I: 99) (8. )  = 0
1t
Z = lv:q, et siet
(F :99) XV
Sumamus jam problema aliquo d, quale est maximam vel
minimam ducere ex puncto dato  ad curvam sonicam AY
quae so  determinata sit MY. Ex Magatur
normalis MN i axem D sit AN, n et MN, m
ipsa autem MY. ipsi Axi sccurrat in P. Ob
2bX +(b: e) XX = jy fit differentiando  b + 1b: C)X, G: y = dy: dx = X B: y
Ergo X P = b + cb. 2) X jam MN: N P = y :, b 4(b: C) X
Ergo sit m. A K + b +(be + (bF er: 2)X  - n =. y:, b + (b: C)X. sit b - n = 7
Sit bre = B fin M b + mb: 1) X = ) By + (B: C) xy.
et pro agponendo +v: q et pro X ponendo, Z- (1: 4)V
fict mb F mb.: e Z - mib: esl. qlv = Ht. q)V + (B: est: 9) ZV - (B: C) (+: 9) l: 9) BV
seu (BFl: equ) vv - (et: e9) ZV - rbl: eq) V + meb: 2) Z + db = B- mb
y Ct: 4
Multiplicetur aequ. Q per + f st aequ. 0) per - mb
_ et aeq. D per arbitrariam dtC
productum
ex aequationibus  et D addantur in unum,
et compositum identificetur product ex aequed
consentiunt utrobique termini absoluti, nam utrobique
omnic aequabuntur ipsi = ff mb superest ut aequentur
aediaquentur utrobique termini respectiri, qui sunt numero quinque
Opas est ergo aequationibus quinque, habentur auter habentur autem arbitrariae c, 9, f, l.t, sed haesunt
Kuntum alunio ut a.ue pua p t quae mo ti plicadit psam aequ. D. quo s ha quionque un dofitiant sut lanm sperot
Crsiranam mullioqus tartanti
mods quibrevor ante tendano