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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 2c
# Folio 14r, page id 00068199:0027
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 aequala data 100 + 110y + 101 dy 1120 yy + 111 ydy = 0
Assumatur pP + 9do + r =o faciendo v = myus + ny, et fiet
pv + q. Lo + Y = n r + npy + mpyys
 100t + 110ty + 120 +yy + 149)+
ser y. fit r = 100t. Per ydy fit q = 11t: 2m Et per dy fit q = 101t: n
quos duos valores aequando fit 1IF: 2m = 101 t: n seu 1In = 2. 101m et n = 2. 101M : 111
Ob y fit  np + gdn = 10t et quia q= mt 2m fiet 1o1t:n, fiet nnp + 101F dn = 1o tn
seu t = nup:, non = 101 dn et ob yy fit mp + qdm = 120 F _ et quia g = 111 t: 2m, fiet
2mmp + in tdm = 2 120t m et t = 2mmp:, 2. 120m = W am et duos ipsius t valore,
quem valorem ex priore valore factum aequando posterion, fiet evanescet mn
feliciter et fiet 2. IO(3m - 2. 101 m dβ = 2. 101 βdm = 2. 120 (3β, - 11I ( (3 am,
et fiet  120 ββ + 101 dβ - 110 (3, i, 111 BB = 2. 101 β = d m: 2 m
sit data aequatio no + tordus + regy. Et habetur M
sit aequatio data -aD Xxdx + gydx + bg cydu = aadus = , et fiet
(3= = ∞ Aa ~C et 100 = XX dx et 110 = o et 101 = - aa et 120 = dX et 11 = c
) Z)C (- dx 10
etfm im) = 4 sC qg (4. 3C) X + f. a ubi omnia puanescunt. Ergo fiat 12O = C1 (cta
 sit aequatio data Xx ak +GB 12O gy ax + cydu - aadus = 0 fint B  2aa c ut ante
Geraliter . dB constans, fiest
amt : 2m = 120, A a dx , F 12Oβ( 178. 4, 10L +I d X :, AI. T. 101. 10t = 2. 161. 2. 101. 111
et dm:m = 120. A dx:, C 14 2( aa) (daq 1ED
120 BB 120 (4. 101 101 : 111. 11) + 101. 2 d (11: 11) = 10. 2 (101: 11), 4. 101 (101 : 111) = 2. 101 (2. 101 : 111)
et fiet Lm: 2m  120. dx: 11. 0, quod non procedit
fit ax dan: 2m = 120  fit sed ita et m et am evanescit, nec
cm: 2m =
obtinetur quaesitum, nisi in casu speciali, quem praestare in arbitr, non est
Ex aeq. vata est  dy = 100 + 110y + 1209y ; 101 + 1y  = (ex quaesita Y+ n p y+ mpyy qutrquis
fiet 100gn + 110gn y = 120gn yy
qany +gamjus
~ 101Y + 101 n. py + 1oimp gy t
+1019dn y + 101 gdim gy
FiIr y + 11 n B yy + nc mp y
11 9dm..
1119 dn 7y
Rursus ex gy fit  P = (101 - 1) gdm = 111gan, :,  (1 - 101) M = M1N, et evanescit q quando
ap), 5 = 100 gn: 101. Ex yfit p = 2. 1209m = 111gdm: 1IM
fit
72
fiat n = (3m, ubi (3 cognita et fiet + = BBmms:, 110 Bm = 101 m d B - 101 Bam
one valore
+2 gutus +290gm yy + . 1204 m y
Ex y sit n ps H0 qu + 2 100 gM  +1 n
p = 10gn + 2. 100gm = 101 qan = (100. 11 : 101) qn, : 101n, ubi rursus avanescit q
valores duos ipsius p.
106 _ _ 10 n7 129y + 11y + 10 =0 differentiando fiet:
2. H2 yy der + 2. 12 9dg + N +y du _ + 11dy +10 = fit dy = yy d12 +  ty d11. y + 10, 2, 2129. + 11
fr
sit data aequatio  Xxdx + gydx - aa dy, seu us = xxdx + qydx, : aa
7y d12  2. 12XX dy - 2. 12 y  H  y _ - IIXX - 11 yy = 0
Multiplcetur  data per +my+n, fiet
10
+ du1 V
CAX
my +n fiet vmy + umyy + 1omys
 assumla per
+ 12n Gy + 1n y + 10n
mulha licetur et prior per ut m, et sact   L um = um 12n seu C - L mdio
n: M  = - 1 11. 12 Et 42. 12 8. d1p = 10m -   (11: 2. 12) m tandem + Xet 9 = 10n
seu L .tXx + d10 = = (I1: 2. 12) m. fit m = 1XX - d1o,:  10. (11:2. 12) = - 2. 12 + d1, 10. 1
2
fint =id12m = (= HIm) == () HIm + 12n Et + 12mXX - L dum = 10 m + n
et
tandem + = IL mXX -  d10 m = IOn assumantur p +G1E +Y dt2
et 20 + 11. 10 + 32d10 = 0 et 40 + et 1. H 4. 42 dy et 50 + 51 12 + 52 d,2 =0
tollantur ope harum 6 aeqy. primum deo, d du, de2. deinde habebuntur ex tribus
reliquis oplores ipsarum 10, u, 12, qui destuantur, prodibunt minimum 6aeqque. quae dabunt
quaesitum