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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folio 285v, page id 00068196:0566
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+y+Y
Assumatir
Aequatio hujus gradu, interpretando V per Y +Y potest reduci potest ad cunc forman 100 + 101 dy + 1119dy Zo
200 + 210y + 2209y + 201 dy + 211 y dy =0. seu -ly = 200 + 2r0y + 22477,:, 201 + 211y = 105:, 101 + 11y. Unde per gucem multiplicando fib
. 200.  u1. 210 gy + 11I. 220 y (4 = 0 Compendio 30, + 31y Aay ar o poho tad 3 1s tod 3r gy d 31 y 165. =, rocon ao noldus rao agan
= 100. 201 -100 2 11 G
+ 101. 200 +1p. 2, 8. 4 101. 220 9y
a31du f  3 2ydy 1 3 3 Eydus apt.  + A. ao y 4 qu 1ol guys ra1. 1 qyslyse
60)
quan
3. = c  L B. 200 y 1 10 v0 rfes 4 10. 220
tutiando ot y) sp P fiat 220 L (o0. 10) (1n dro +2ro du1) et habatur 220 nssob guy dy, fiet, 41 = 3. BB. 33 + 10. 21. ob gy siet 19. 10 + 00 d32 o de seu 19 20 + d (1 O1 o) F M dno r 20dn - 8
O. 20. yay + 19. 211 yydy N
cui addatur assumen 2 mulipicatper 10
et habetur 210 c  B yty eb A Gtu fit C 3 00 33 10t + 10. 2H tor ob Idy fit 40 et ty 40. 111 =, 00. 32 +10. 201 ) en 40 M - 2 _ set 220 + 5 11 20 f 10. 20
220, et, 210, datur et 32
itaque ut aC simpliciter, peducitur ad 2II, ita 40 simpliciter reducitur ad 201 vb du fit 00. 31 = 40. 101 sel gbe in seu . 100. 2n + d roneto L  M. 100 - 110 10L
n
db y fit 41. 100  10. 200 +00 d31 seu 3, 2433 100 +10. L11. 100= vo tv. 200
den
sun - 100. 1II. 211 (F 101 III. 2102, 11. I1I. 200 LD (per 1) 2. 101. 101. 220 ( (2) 101. IIL. 218 + 10, 101 201
Ly 300. 33. 100 1 10 2 100 = 00 10 200 + 50 tEm rud  avdua het av iend 2I idurroo 1 udevo
irnius ob absolutum sit co  cord siu 4e = 00. 111 d30
no reducta A ad aeqqe. 1, 5, t7.
ob su 00. 3 ex. 11I. 3) 2 101. 32 +10. 201 ob y fit 41. 100 ) 10. 200 + 00d31 seu 3 50b. 33 10 et 300100. 33 +10. 100. 211 = 00. 10. 200 + 000b d31
seu 2. 100. 101. 120 , 2. 1M 110  1 100 nol d 1 dioo Eot. D 1H 100 d20G + H1d1o1. 200 + 11. 1et d200
B comprutio sit 21ot.I cao +dt M NO= eo b 57D et fut 2 et 00 H: 101  S1O et nM: 10  Si et fiet 15. 20  = 510. 21 + 5i 200 et 530
e
Deniu al termtnum absolulum est 46. 100  00d30. seu 40 +00 40. 115100 = 00. 111 d30 seu 2.00. 100. 32 + 10. 100. 20=00. 11 d30
E 32 31 et 30. Ex aequ. 17 fib 2, 00. 100. 32 + 10. 100. 201 Z a N. 1 1. 4100. 201 - 111. 100 1 201 = 111 duo1. 200 + 111. 101 d200
et fiet
superest jam ut explicemus
ubi cum 32 sit cognita,
200, 100. 32 4 (10. 100 - 11 d100) (111. 31 = 2. 101. 32) = 11AB C M1a 31 + d1. 31 = 2 d (10131)) = 11 d101 200 +1.. 101 d200 it habemus binas aequationes 15 2t 181
faciamus 100. 10. 100 201 = Utdioo 201 1g. 100 d20 cujus apo toltatur Et tolatur hinc 201, ope aequ. B3
in quibus explicando 31 _ habemus binas aequationes differentiales simplices ad binas literas 200, et 2 21I. jam sublato 211, sa redutatur
res, ad 200 2.. 200 +. d200 +.. dd 203, et fiat ad200 = 0: et res reducatur ad 200, 1200 ed quia aderit simul ddx, nonideo ad exitum plavenire
dy ytus sit y - B b v sit dy e v do gdu -  d sit e- 1 = 2e 1 fint P = 0. Huc ergr non licet
sit y - bed Idus - B deo dua b da _ quid s g ( y- ero syta ipdco y yt  = Y cocot   codco-
sit 1012= ton d (101Z) = 100Z + s 111Z = 1oidz + Z d10
101 100 + cor y + = 10 + ny 1 4. du 100 + 12ogg + 10 ly = R 100y S 100 + 120y9 + 101 dy y = Y 10 + 12v + 101 e V av
100Z + 101Z dy + 111Z3
100Zy + 101 Z ydy + 11 Zyy - 0 sit 4(101 Z. 2) -
licebit
i co dco +  da 1covt d t co d 
data aequatio 100 +  10lby + 1 ydys =o multipliaetur per 20 +ry fiat 100. 30 + 181. 20dy + 111 20 ydy + 100. 21y_ + M yydus = 0
A Gtu fiet et 3.33.U ob gy fit _ 403- + 005 = 8 3
Et psi aep. 3 disferentiatae addatm ipsu aeq.3. Multiplicatar pe dodus
ione constant
a  d (Lyy) = 4G est 9y it + Aitylys
Assunatur  30 +31y + 3r gy +33y =0 et differentiantos, 430 +. gldy + 2. 319
1i(32 + 1. 13 133)
lui addatur 4i 30 + 31y +32 9y +33 y
+ 101. 21 y dys
Et ob _tque g.oetum
(um absuit _33, Y tot 40 = eod33. 33 = 00 d32. 32, quae aqua Ergo d 2 2 et 33 sunt ad in D.ca
tante
non licet; nisi et sit determinatio in datis
H1. Et ob  habetur 20 = 2. 00. 3
NG +11. 2yy dy
101. 21 y dy +100. 2
3.00. 33,+42. 92. 11
20 + 211 + 22gy, fint
ita binos tesus 21 valores aequando 130, 11, 32 ,e3 si taater et inui lavicem habent relationem simplicem ob f fit JtG3: 3i = ao sin sig. 1 X E1 tatm B quod 
An datu multiplicetur per
pona
0
possumss lotere si
oltiento 30, ustan 32 e 31, nam 30 jnatulatur. 1 t. 3. rdolitub ad S1 It rabmus A epu. lui quibus htm H1. H ast prmose tndo 31 . 1, haletum 4 2per 40
mutienenta per 4 4b + 42 ay,  0.  A0. 30 440. 3t nt tu B2. +av. st . 4 g2. B, 4 grigsi . 4 r. 32. . gr. 32 fo l9
habebitur 32 per 3y et 3 per y, et
hatumu hatemus binas a pe quadum 
Quod si sic.
1. 101
111 - 42. 30. 111, - 101. 101
00.32t01 + 42.31. 101 2.  a g0 ut datas
Tob ge. co ui pt 1 = -
a
obsolulum
aliunde jam habotur cato
eet u di + (01 _dy C0 fiat
nesc
in duas, ut ero
utr, 30, 31, 2
quids aequatdone
_ F = +W =0 = Y ydo -vau  gy duo  t auluis igo talis aequatio solvi potest
sit aequatio data Y 5y + g = fyyAgu + hdys = o multiplicetur per Z. fit fZet szgy + g gzy + tidus  0 fiet lro = h, A 1 = stidx
bata,
tificari
du = - fv: h
Habentur ergo ∞ et W, adeoque omnis aequatio syis +gy + hdy =0 solvi potest. Hac est insignis promatio calculi nostri
a  + W + Cy stgrey  fdg r yds (0 sit ydv - vay + Cb 9ydco +  + c t f jg = 0 Hini si fiat duo - - ctf. 5 soo W = 5as )
s
prodibit ydo - vdy + cytus = 0 aequatio situbilis. Itaque et haec est solubilis de - vdus + cdy =0 ridiamus ergo an gereralis
f. + fdy + hd possit reddi solubilis
Hep +Wd + Cs
y
n
sed non apparet
=0 fit + ydo + 2so + vdy + vdip + d gydwo + cydys = 0 li absit qus int W constans
 I
ipdu
42 ydlo
+cydep
+du
apdy + epde
prod si jam  absit etiam y, siel du = - ct p. sin v- et 4
Y  _ Eyytaitus a ngrsez voq  gtae Avaqu sit Zedo. g = v. h et (av iv = sta g h et v- b ( 5
sit jam aequatio data f + gdys + cydus ∞D sed video non sufficere data nisi praeformetur.
dyA s
ea 
+Z= b(5.. ). h En. f(gy + hay, b(f.. J 2b) = y. b (
(Ebr g Fg 20 sit ly +7d5+ figduy - 0  ino h r7 solibile
Sgylus
Soyr yes olyr eq . k 20 fit Hlyn 9tf. + fg  syplgs r fydh 20
sit data aequatio: rm f angus erpdy argydus = o mulliplicata per arbitrarum r. ft r- 5g. m 2t ro -5 seu r = f. p ob gysit r= fagin et habitur 8g = n. p
fm + lu  = h n- hdn stratque adeoque gest inventa, ob ydy fit r= fg: 9 - f. p. ethabetur g - 9. p. Ergo in datis fit d (9. p) = n : p _andim fit ob 9 fit df + fah = (rm) fM. p
si t se0
sit n - c fit l0 +
site ) ndi patet hic A esse superfluaum et tamen sufficere arbitrariis. Anergo sic sIy +9 15 +de a +gy + h =0 et fiet
Syt e
= fdys ayds + de  fgyay lesu +fdygy _ fb Vittrnus a rngus + ip dys 1 rq ylys a  6 9b est rq utg seu rufgique se ypte r segin
sit leg Cd =V C0- V
 produ 16 + f =  du
seu l. b s - hay b
eg +edgs redhi
lon L = s. ded 10co ts 4 hdleo
+ fahys
Et ( lo: 4 = q: H et habetur g, habetiu et r. f
ob yd est ih = r( = de, : e = (ob y) rm - d, = edg 3. f sa H. rea . B. g)  B p. g er mr rd cu ) 
ob ly est f +eg = rp et hobr habetur et C: f Nom f= Yrido) sit r. tus )reg rp, et ed  B nitol -po:g
Supersunt termini y et p.
sed ita utrobius destruitur dr  adeoque evanescit et Y
Alitr fty (segi 345 + de a cf 9 hd + 3sg + dh
5y+e.
fy+h
+ K -o fil i fgytus + ga) yy Agyde + hsty +hde
Halemus quotuor aegus. et tullendo E et (dk. resabunt duae fl. 9.
pro quibus adsunt literae 5 8 R e, f, g, h, quae tra lantur eodem
hydf
esgydus afdg gy nsus ih +egdus +eedh
Weydu
+fgd kpy Hary E ehdk
=P, : 9
= - edg 
teim dale ponandis abasse ydy.
g
ydo
V el +rmyarngus + rpaya sit   fit c--
Et d  ( = h.f - 8g,: p = ch ik
ob d fit
modo exciptis sigms.
Y 10Y + A dy +rEddu = 0 sit y = b fict 10b + 1deo b + 12 ddio b + 12 daedeo ( b = 0 sen 10 + 1i deo At 12delco + 12 daodeo = 0
+W =0 fit puie y A + WA - vsus + yydu A V rL + rmy arn gy + rodus vagr tuo- en dua rap - plr - ob gu sit du = go rn  = - rp do-   ptr
yd u
a fio pubre vortur votasitorsi sti toit
rdp +pdr + abrn = tm et fit P = m + d p) - p E y : 2B
sed ita calculus fit difficil
e co  100 + 10y + 12099 + 101 dy 20 sit g- f P, tit 10o q uo  toss t10r fil +101 Za f. = 0 inq faci coci potest Z
= G sit 1o su y = p EF 531 d fit 10027 + 10532 +u0 Z9 + 120 ff22 + 2120fg2 + 120994 101 Z 2 101 22 t 1 Za- 101 sy a Z) gdq
 vuse utus eroisieis in geta qut si qua une tatum quarius aeie tani ottam plae puon sue petumi suones insunte ospuie ra e ogse  uste ro pretutum
120422 +n 2 +1218