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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 2c
# Folio 192v, page id 00068199:0384
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1. X
Y =X a. X
X
ax logx et dy= dy: y -d x chc. Y + c elque ax : x + alogX
et d X = ux a. X, Y , 1 + logx
logym aX lyx dy: y -b xs  elogX Eb (E + hogx)
C)
d(xax) = ade Xa  XbiB, 1 loga + bt ) L+ bX (log X F bg L log X)
c.X
y _ x. X et  dy. y - lX a, X
A 14bZC X
1, Z4b s -ax A es  a tat X logx + P. X  X. bogelogs t abe X X. u X
y= X .X dy  d (X2 L) = dx, X  ,. X,    a + elg a. X. l0gx
a y=
pa.XBa
(aq
CglB -  pdy: y = a. X LX dH. X +logX (x  X b 4 7 , + 1 log X)
b. XC. L
t)
= dx p axb C b.X a + abtilog X + ab. 1. K X L logX loge
dg: 9 = dx, Xb. X, a + ab X  + a bc X. X. logxlogx + abc. 1 X X. X. log, 
axlx
e
x
ax
X
a
c
=  Xbrba
Hic autam a sunt constantes.
Ex  X
d
id videamus jam si variantos per omnia et indefiniti sunt exponentes
y- ZV log y= lL n lgx et dy: y , m. X +
ex   ex
y = q ly = lm. n dy. y =pdm: m + duum, sit d m = m, n  drlim) i  dn lm
y= M ly = n Im dy. y = amE: m) n + lm R n (dm  du (:. 1) p + dp Ln)
et d (m. ) = m. n () dn. + m. n. fa lm du  + n n p  lmln de
Ee a () Cp, n oo + n pi lacm + pe ptutaq laba de 7 p plai i itup i qua p P olapui 4
Et ita 2tiam in magis productis
Haec secies comparetur datae dy = b dx + CXdx + C XX x +5X AX + q2 dx erc.
e
sit X.M seu seu X logm ge
Et sit ry: loqm dua n = bdx ( sit dn X, tict N, =  l = b
M
Et bx p lx dn = ex Ex et p tu = ex dxo: bx l d
ret
sed npq i = p In = log 1b + lX, et fiet 1b + IX : 12) dn :n = cxdx: bx lx bX (x
M
uge
seu fiet dn(: nln) seu Un = (, c∞dx: bX +b H (X, beb 1b+ IX
Sed si fecissemus X = log  m, seu m - h siriet n = b sed hoc un licet quin sic ofieet y= log H b
e.
imo iideo nec hoc convenire i ut p
8
1a + abX log.x + abe X. X. loga tabidX. XD. X. Co)p tcC
m
satim habeatur velut per bx, ita enim slatim
et y habobitur, nec opus ert progressu iar
upt pulnde teu
=11 F 6 praestut ergo sic conferce dy = m, n du + n pl Alm n + n p
peu u
et
Et  et sit m n dm = bdx et dm: m = dx et m n = ab et hs logn + t pr hogn - log b
 seu n X + logn) = log b, ubi ZX + bogZ = dato constanti, quaeritur Z, sed ita
quare non liceent facere
etiam statum finitur inquisitio, nec opus est prigredi in serre m d = bdque semp  en = b dx: dim
arbitraviam suppositionem sed manebit. ne a
et nod pn p
eta
Imda = cxdx sen Vdx: dm) m. pd lm an = CX dx
Mek
tc
sen. b(dx: dim) pI Im _ dln = cX dx Ergo p =, (Cc: b) (X: 1M) (d X : d ln) dlm: da
sin pa = (c: b) (X: Im) (dlm: dln). Sed ita misus finitur progressio nec opus
est ultra conferre. Sed ita in universum patet tales serres cum
ad Comparationes non inservire
a finitas, ex guili ZX + logZ = log b
sed res utiliter serviet ad
sit  y = M = bX + cx fi et
log 7 lm = lb + ta ly = n Zam(m) -+ dz lm) = b. x + 2CXdx. hanc aequavem
dibidamus in duas ut sit  Zadm = b dx et i idZ ln fiet ZEdm: dyln = - b. 2CX
et m = bX + CXX
seu 2CX: b = seu dlogus: d loglogm = ECX: b
E quae duae aequationes defindunt m et Z quo posteriore Z logm = _ log, + log (b + CX)
E Et
dineruliter, sint duo termini summandi _V Vdx t dx sit quantitas summanda
X dx Vaa = XX) sit dx = Zdm: m  +. dZIm et fiet X - Zlnh et M = saa - XX), su hoc
eno sit
idio ja non licere t  a X) et 
Aeni 2) = m d loi i asp. n f 1 rd 
(37
5iA.
7+e02
st
diu
M