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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folio 193r, page id 00068196:0381
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i
Aadus + b RCy +C
26 septembr. 1693.
sit aequatio resolvenda differentialis.
squam mihi hoc mense proposult Hugenius s haec erat: aadus + xydys
+ X9dX + ay dx = axdx Hanc transumtemque in generaltorem, ob rationes
singulares calculi relationum, nempe ab du +nxydy + Xydx + cydx = axdx
ubi n est numerus, sed a, b, c, sunt ipsis rectae.
Cydx = C, 10 + UX + 12 X +13X + 14X +15X etc
IXdx =
10 +1
12
13 14
Abody = bb, 1. 11 2. 12 3.13 414 5.15 6.6
N Xydy = n, 10.t.A 10. L. 12 1a 3. B B. L 4 10. 515 10b. 16
nxydy = n,
11. H.Ht H. 2. 12 1. 3B3 A 4 H. 5+5
R. 11I nan 13 B 1. 41
13 L.HI. 13. 2 12. 13. 313
10. 1. 11 10. 2.12 10. 3.13
11. L. II. 11. 2. 12
13. 1. 11
sdedum Dari possis talis explicatio ipsius 1o at cxplicatio
erg non crlsun t finita; vel se non st fitula, ut
-a
=aXdx = 
14. uutt 14. 3 12
+i
10.9. 14 10,5. 15
11. 3.13 11. 4, 14
12. 2.12 1.3.13
13. 1. 11.
14. 1. 11
10, 4, 12, bb, c, n et X
20 -21 X- 27e
net reti
faciamus y=, 10 + iX + 12 X + 13 X, et. Et fet
dy = 11, 4 2. 1 EX +3. 13X 4 4. 1 X etc
fiet 10c + 1. 11bb = 0 seu
fiet n= - 10c: bb
nC + 10 + 2. 12bb + 10k un - a = 0
seu - 10cC + 10bb +2. 13- 1000 ena =0
seu 12= 10CF- 10bb+10 10 cn + ba
262
Et ita porro reliquae literae, ut
13, 14 1s, etc. inveniri possunt.
verum si abfuisset= axdx, idem fuisset
calculus, eu habe excepta differentra
in valore lipsius 12. Nempe loco yponamus v et
Nempe fiat CVdx+ vxde +bbdv+nXVio = 0
10, 2G B bb, C, 2, X. N uum de patF Et sit B = 20 + 21X + 22X  + 23X + 24X  es = 0. patet
vel anstie llicemrus: y - 10 NX - 1 2X eddem modartaberieque Ht rur Het
 a
p coruu
taldrem
 mi haberi 23, 24,25, 26, et c. eodem modo ex , , C, b3, n, 20, 21, 22
o  13, 14, 15, 16 eter habetur ex C, b n, 10, 11, 12
 ine potro sequitur  eodem modo fieri ex X, C, b, n 20; et 22 sumto pro 10 cc - 1obb 41010 Cn; quo
ill. ea
u in sebiur D habene
es
b
y fit ex X, C, b, n, 10, et 12 sumto pro 10cC = 10bb + 10. 10 cn - ba. Quid si ergo sic procedamus?
ope aeq. 8 tollatur 10, seu vulor ipsius Et ita habebitur V ex X, C, b, n, et 12; eodem modo
ut  habetur ex X, C, b, n   12 - E. posi posito 12 esse arbitrariam seu supernu-
merariam, quam licet assumere pro arbitrio.
Sed eodem prorsus modo habebitur
V ex X, C, b, n, 22,  quia idem prorsus est calculus. Nam si pro
Nam si ubique pro 10, 1, , 13, 14, etc substibuas 20, 21, 238, 24, etc. et pro
2. 12bb = a sub stituas. 22bb idem est prorsus calculus utrobique.
Quare si
soluta sit aequatio 9 sic ut reperiatur V, ex X, C, b, n, 22; etiam habebitur
Y ex X et datis, nam tantum opus est pro2. 2e substituere 2. 12bb -a. Tuomodo autem
20 vel 22 ex ag. 9 solvi soluta haberi possit  ipsae series rursus occuurunt. Na
redricta et cum nostra
ratiocinam
dica
i 2
. y
∞UX.
et abb- a, ut
Nunc igitur perinde est, ac si rem redegissemus ad lineam qdaerendam quae
desiruetur aequatione Cvax + Vxdx + bb dv + nXV dv. 2o Et ten jam quaeramuse
 per X, quia tunc iterum rescindi potest bb dx. Ubi vidi posito  3 4 X x = 30 + 31 V +32V etc
non dari exitum  aeq. 9 per seriem explicata. Rursus si X = _ + 31 +31 Mt. datur quidem
exitus in aeq. 9. sed non datur si emitteretur ob du, ut expertus sum Ergo fiat, X= 29V+ 30 + 31 +3
fiet Ax= 29 X  - 1. 31 - 2 72 eH.
sn Vxdo = n, Au vB + 30V + 31 + 3
CD dx = c,
+ bbdv =
WXdx =
Nunc ergo explicationem aequationisu instituamus
+3
+ 34
X +29 X - 31 - 232 - 3.33
+2929
X +bb
X
X - 1B1 - 22
.383 -
+2930 X - 1.303) - 2. 3031 -3. 2. 3
+29.31 X
29. 32
- 1. 31.31 -2.31. 9
X - 1. 31.3
29. 33
X
29. 34
et fiet a 29n + 29. 29 20 seu 29 = - n, et
Ed -C 3On - en - 30n 20 se-
3 C sed ita video fore c=0
quod est absurdum. Et ita non
procedit haec explicatio ipsius
x quae adhibita est in aeq. 11
itaque oportet in aq. 14 ponere 29 =0
et per _30 =0 ubi res procedit si
exset bb, sed si absit bb, omnia
evansscent. Ergo ae9. 9. hac
methodo sic trassen  non pt
ut ulterius reducatur