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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 2c
# Folio 95r, page id 00068199:0189
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Haec videntur succedere
0
sit aequatio data f+gyt hdus = o siat m+ny =o et differentiandus dm ++ ndg seu
seu logn- seg:  et n = b .  den
am: n Hdn: n) y + dy = 0 ~ f: h fg. h) y + dy sint dn: n = g. h (et m = s(n f: h), seu m = (( 3 , f. ti)
et sit y = se - s (bsd: , sih): 119.h posito F+ gy + h du = 0 si 5=0 tit  dy. y = - 9. s et hgy - : ( q. brs
data
seu y - 1:, B  et y.
dacitur, us observes orine s tiare co
e
atem euto licet sulmitadun omenu a
1p. B. 4 cg. 1
Sit f + gy +hdys + Kyy + lydy =0 Aessumatur m+ ny + pyus =0, difeaentiando dn tans anas p  y+ jusdus = 0
inttoplocatae, per 9
cui addatur assumta multoplicata perr, et quod prodit coincidentietur datae, (biet
am + duy F ndy + dpyy + 2phy dy ∞ fgt gis + heluys + Kguys + l4 dus
mr +mu +pruus
per ydy fit p = 19. 2 dytr = hq per gus fit r = dp  k9: p - dpip = kq  yl qu2) - ap. p
seu r = 2k: l = dp: p iper y fit r = gq: n - dn: n seu 9G: h(9) - dn : n
Ergo  2K : l = dp . p = g: h = dn: n. Ergo s (ck: 1 - g.h.) = log p = logn = = loglq - loghq
= L logl = L logq = logh = logq seu -25 (2t. 1  - g. h) + logl - logh = l0g g.
seu
q = l: hih num (52K. 1 - 9. h) seu q= l :hitr B e . t. ) p = l: , 2 ht . 1 3 i = 1.: M B ..
r=  g: h - a hq) . hq
ontandem am +rm = fq, fietque tandem
O= dm +  (g: h = et h q. h9) m = fl. tih bd adeoque per priora
P(( sim -= dcndtibus, fB ) posito q = l: ht b S. 19 - g t.
Unde tandem B - = lq Vi + hq Y + S(b s Bh)  aetq) ica B, 54) (Se d. e existen tes= l:, hh 6 Srt. 1) - get
posse abesse l, quin simul absit H; nec potest abesse M
in hoc calculo non
sin s aostt.  facilem Terminus  suppleri potest faciendo y v  v = yt e
possum
quia simul absit g.
potuisset etiam assumi in +  + sp du+ gapt d = ofiet ram + rany + rndy + pdn gy + pnydus = 0 si n constans, abest qu
Si ab sint 9 vel K sit nocet
amus 
qui si sic n A + seagigenelo   er =0 fiat du E + Peyng eptar = edl yr rAt - t gex d. Apius B. 1 dr
de +dp y + pdu
tit am pregus + ampry + rram F ppydus + Prydp + py ar + rpdy trpy + rdr = patys +  rdt- topy.-t pay - + d
pytri
qu (pytV
sed si velimus evaniscere ydus, faciemus im + tr -o fil da + (tyg o) depyuo) - pg r) A(I3t 10) = 0
ubi evanes cet gy ponendo dm = - dp: p
ou am +2ot dmy
War
-rduo
tytio
Ltlamyys
+W ap y +Wpdus + + ap yy (+ tpydus =G
+ drty
- Y at y - r Fdy - pat gy (- pty dy)
= duopus
sed sic calculus paulo per plexior
Interim miror haec proverure ex im + PytY -0
unde int  tale derivari videtur
Fytiv
dm x = , su = kg, - d (4)
au
ob y et y7 fit Fu lio) = K 9 - d(B)
N
ituque dm =
qu (+9 + 10)
~ fit gp + hisus etequs
Capui
onx- rtK
cq= Wd rF,ih = H dm + dp - pdt,: K fit dm -  spbr .
Et Y = LCL  Krop - Hkam = ht dp + hpdta: tK
WWp - rt, k = cuwam + Wd dr- Yduo,: 5.
et din = wops. xts = Warh +rdcoh,. WW
apra dm = copsC rtk taph + path,. H
(_  F -b9 -1
rd + men
coincientio  e insitenis.
unde
superest rdm + uodp - paro + tar. rat = 34 si 2o = t fit 2dm + d (f) + d (F)- 3
fq - tdr trat 3 Kg- t ap + p at et q ∞dt+ t ix B +t an A. a1) +d q ) t de- pat - tartrat:, K- f
= h9 -1 seu 9) +p- tr, ih 2dm = 34 _ - t dx rpdt - tartrat, quem talorem  o
U; de
 superest ut conjungantur duo valores ipsius 4, nempe (3) et 7) Ex is fit Abdqt qth = t dp + pt - Fdr - rat
2
fit Epger, Ks  = Ep. pat =tartrat = Cr, k, 5. seu a (f v, 1) = K. f
et prius per 4 se erat (R - f) 9 = + dp = pdt tar + rat
et dabitur pax. t, adeoque habebitur quaesitum; seu log (p - r, : t) =
r3: t
(K- F
seu p-Y-
L S(X - f, ot)
Resumanus certitudinis causa) iuta est aeqratio F+gy + hdy + Kyy = o assumatur in + Pytr = 0 sit dn = teu + eF glys 1byer
gtuo
eot qui neoi s ande
Et ordinando idm Wulo + rducotu + Wopdus + dn E 4y  f9. +913 + hq dus tKgyy = 0
+W dr + t d
= r du
AW ap
vol ram = B
lu
pmtor
s
- rtdus + tap yy -
ex s fit am = t - d∞ = (ex 77) K4 - d 4)
Hax-rat = 1 dp - vdlu  ex dudbus rtes 2dm = 58 +  K4 - d7 - d P
- pat yu
Exysis idm = d2
ad
ret tarerat- iopapduo prioubus su
quod obtineri potest per aequationem
porro si Wo= t, fit dm = 5  - d 7 = K a -) d F. Ergo q=  p +t d (B: t) : K-5
LaZd seu ram to modo sit in datis d = f+K
Ergo si + = W, coinudent bini valores ipsius 2 dr
superest jam ob du, aequatio, wop - rt = h9, seu tp - rt = hq seu  e P= = ht et fit 9 = t+, (p. B: L): h, et duos
ordinariam, faciendo v = y + e
ipsius qualores aequando fit  d (p - r, : t) : (P - Y, : +) = h: R-f ut ante.
Rem tenteuus in exemplo sit XX dp + ∞0 = 0 faciamus v = eHy et firt t xx - dy + 7e y tiet f = XX det ee g = re h = 1 K = 1
tam delet ess frK = 9 fiet xX - de + ee = 2e, sed ita hoc praestare st rebere ad eandem difficultatem ergo si adidaamus v = et cy
et f+K 29 fiet XX de + 0C + cC = 28- dc. et fint = de =-de seu et e = c fint XX + 2CC = 2C( sey ee - 2C + 1 = 1 - = XX
fiet c 14 L1 (2. 2XX)
ed sne suppositione Wt fiet um = Es
pursus me
34 - L 47 - 18 d F. = 97 tW - trot dr + trora t L lodp + t lopdu
quro
mescit C, ergo ala
drodotensq u
stitum siimantra te pos euo s tie toiti or ie o polorndo a simperuo
loga-
urH.
2H2 -aE