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# Title: Leibniz-Handschriften zur Mathematik LH 35, 6, 27
# Folio 8r, page id 00068072:0015
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de est sucessus
10 Novembi. 1703
Multis dentatis transformationibus faciibus, quibus tolleretur litera
parameter ex aequatione langentiali; plerasque omnes comperi repugnare
naturae: Una tamen novissime in mentem venit, quam nondum refutavius ea
ob rei momentum examinanda est. Alibi ostendi tant Aequatione
Jangentiali dari ag. dx ex y. a, et x: a, posito X et y esse cordinatas
et a parametrum. sit jam y: a = deo: do et sit x: a = (2d0) Sddv:dZ. ubi
pono lco esse constantem. Ex 1 fit dy: a = (4to)  ddx deo: dodo et dx. a = (5to) ddv: dZ
erit dd dy: dx = (6to) dZ deo dodo, sit jam aequatio data du: du  (o) extgu,: ac
quam explicando per 6, 2, 1 fiet  rdco  qu sade + deo deo seu Sddo 2 szdco- duodeo)
seu adv 60) deo ddze
avdo
dcodv ddZe = 2 (d Z da = deo des) ddv.. jam est in nostra potestate assumere
2 dv S Zdco - deo d co)
i F valorem ipsius da, quem permittunt Leges homogeneorum infinitesimales, et compendii
2d0 do Y(dZ de -deo deo)
seu dav-
= Causa esto dzdeo - deo dco = deo deo ddz ddz: ddvddr fiet dz - deo = deo duzaldz: ddv dd
seu dz dav ddv = deo ddv ddv + deo ddZ ddZ, sive de = (13) dZ ddvdiv:  + ddz ddZ
2x Ergoes aeq. 11. fet seu 2 2 dudo ddv razdio -deo dec) = t Zdcodvaz ddz - 2 dZe (d 2deo - d codeo) aldus
: E. F fiet favav dco sadz) =  15) cedvaz laZ - 2 dz deoscleo feli faz: ddo) Et pro deo: ddo in hac
aequatione substituendo valorem ex aequ. 15 fiet + 2dvav = (6) dudz - 2 dz ddque (dZ ddv:, ddvdlro + duz daz)
e Et sive +Lavdv (ddv ddv + ddq la Z) + e d Z lZ dd Z ddv = (17) duaZ (ddv ddo + dd z dd 2
= E sive fred (avaz = 2dv dv) (ddvado + ddz dd Z) = (17) 2 d Z dZ ddz ddv
A 
1 + 1(-
dudz L 2dodo
unde redu
qE+ adz da zue 8) (d Z - 1= ldvaz f.do 1v) ddzi
(2dody+ dodo (2) (avaz + 2d0 do)
d2 dZ + Ae2- adv + 4 dud Zu   avdo d7d7)
dudz +. 2 dudv
4 2
4
x ce ducr
jam constat regula generali, fone a
d Cdv: dx
Datur ergo ddo ex do
et sit compendio dav: ddz 2  rel (av: d2)
dCdv: dZ)
rel (du)-
Res ergo quantum judicari potest redocta est ad quadraburas
E Sed quia calculus polixior assumumue exe videamus an liceat exemplum
rax + ab = gy. fit adx = ydy et _a:y - dy: dx = 2X + b,: y
et a dx + b Xdx = ydy et y: (a +bX) = du: dus seu
 adhibere facilius:
ey = a + b x, dx: dus  jam (a + Bx) X = yy - ax  seu a + b X = yy - ax, :X
FErgo y. = (yy - aX,: X) dx: dy seu Xydy = gy a dy:dx = yy -ax,: xy ita b quod
:2aX + XX b XX = yy
notandum tollitur ex calculo. idem est si ex ope aequs. 1 et 2, voluisseruns tollere B
D. NNam b: a = yy - 2ax, : x() =  ydye = a dx,: (dx et fiet
4y - (2a = ydy = a) seu ut aute yy- ax = dy utrete seu yy - a x =7
de  2Y 201X - 2x dx = X do
av = ax,: xa = dv: 2dX sv. X): (3 v. X) - 2 (= f. X))
sit ea log X + plog b: a) =L (Gy: ax) :, (yy: ax) - 2 =log2y - 2) dco: 1 - 2a Blo 1C0 - a)
a a dx. x = ee
pone yy = av fiet
dv: dx = 2(∞ = X) - 2
log(y 2) log, coeo qdta se XX B -c d log eo -24) seu
gy- ax: xu = ag: de _ sit Y: a = alv. do idx et X: a = Sddv: d2 fiet 1
yg - 24 seu Xxb: a = gy -2ax e ut oportebat
et fit  X log. X B
sed nostra methodo pova
dyidx = (dco ddp: duav
(ado: de seu dyidx = duo dZ: dodx
duo deodu en
devJ: dque = = deo da acodco aZ + dv)ddv des dco du (dcodio daz L
y: dx = (aZ. d) sdav.
s fx): Sddv: di)
-(dco: dv): Sadv: dz
se 2 - a = 
sitsudd ddo Ecar. aa; lodvurc dud. d)a +dav: da Cardinadx tavadus: C (mo ralur. 
Ao . Zd. (Z + 2 d7a v dd vaaq= d Z dZ ddo dv = d vazue dd2 dav +. 4 a  b ddv sdZ
du = dx ddz seu log d
et (i deaqe ddo ddo fa
hypotuem nam ita fo
ib=drao ddo i 1. ded sdaidoteo    1.  omnita ab d
dedo. 
1t .
llodcoZtdd
lcd. Se
dode Ado) : 2) (deo deo de + dx.
povissinque vatore
n Et exirubendo
 iseu ado C20)