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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 2c
# Folio 182r, page id 00068199:0363
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quaeritur
si in data serie infinita datur finita relatio, inte perpetuo vera intem
e
terminos vicinos numero finitis videndum an hinc problema reduci possit ad
aequationem finitam x. Idquidem facile si non nisi differentiae
et summae aequationis datae per x potentiam ab X multiplicatae componantu
sed si duae affectiones ab- x, ut si y + dus + xddy faciant 0. saed
si difficilius est negotium si una affectio insius y in aliam multiplicetur. Sic
v.g. si y = - a + 10 A ux + 12 x + 13∞ 414 x, et.
dy =
xddy
1.H +2. 12X +3.13X +4, 14X3 +5. 15 X ere
6) 12. 12X+2.3. B3X +3, +.14X3 +4515XY etc
) E 3
0
0
Hinc si ex aequ. 2 et
3 quaerumus 13 et 14
habebimu eas ex  11
et aequatio illa differet
ab ea qua 14 invenitur ex 12 et 13
quod numeri Nithmetice proportionales crescunt, itaque peraed
tantum pro +ic H  2B scribemus +a B) +XX B
a2. 12 BF3. 13
11. 2n. 2.3
+nb al20
Tn. n1)b
e-0
. 
quod si velimus etiam n tolli ad hibendo sequentem praeterea aequabionem b + (nH) C + (nnrik-0
tollendae n, habebitur aequatio vere generalis inter a, b, c sed contra si detur
relatio inter a, b, c, an licebit vicissum invenire aequationem differentialem. Hoc esset
quod vellem. Sunc si den Nemee si relatio inter a, b, c; plures aut panciore
sit talis, ut a, b, c assurgant ad potentias, videndum an intermiscendo n, liceat
Eivellere aequationem in duas fac simpliciores, in quibus onta, b, c non
it erdunt ad potentiam, ipso n ad eam oscendente sed oportet eas duas esse
similes inter se, ut quemadmodum in uno se habet n, ita in  altero se habeatn
 hoc modo proprietas differentio differentialis, ex. 9. reducitur ad differentialem, vel
d differentio differentio. differentialis ad differentiotam differentialem.
aeterum si p perprexcora sint omnia si affectiones ipsius y ducantur
invicem,
y = 10 + iX + 12 X + 13 X3 + 14X 1tC
yy = 10.10 + 2.10. 11 X + 2. 10. 12X 2. 10. 13X + 2. 10.14X et Hic perdifficile est, ut efficere
Sane si tullamus
+ P. 11. I1. . + 2. 11. 12. 4 2. 11. 13.
= dio + 2IX + 22X + 29X
0: 15
q
ut relatio sit constans pro
numero terminorum finito
F 1. 12. 12. (in qdidalo pesi relatro sit constans
+ 24 XY erc in y
et pro sequente termin
21 ( 22 = 8. 18. 12 + 1. 2(2: 4. 10 etonegue
sit  = 2EF. E
et 23 = 2. 10. 13 = (8. 10, 21. 22 - 1. 213]]:, 510t
seu 23 = 16. 10. 1 + 8. 16. 21. 22 = 1. 213,: 16
et prosequente 13 = B. 102 23-B - 8. 10. 21. 2 + 21: 1b. 10
24 = 2. 10. 14 +  (8. 15t. 21. 23 - 8. 10. 21, 22. + F 213: 1610):
24 =    32. 10514 + 8. 10, 11023 -9. 15. 21. 21 + 214, : 16. 10
e
Cs 12 et pro sequente termino 14 = 1610524-816+ 21.23+8. 102 21, 22 -21 3
itaquef pro 1d scribam 20. pro 20 scubamus 20. et 10 =20.
gy. = 2010 + 2. 20. 11X + A. 28.12 X + 1646. 13 X  32. 16.14
1. 20.
+ 2421
4.207
+ 9.20. 2. 22
= 2(3
310
+8.+ 2.23
- 8.2027. 22
21
16.20
Ita semper terminus quivis habetur  omnibus praecedentibus
sit X=
10 + 1X + Ve2 X + 13 X
Sit y =.  20 + 2 Y + 22 X + 23 X + 24 (X +  25 X + 25 X
stu 
erit
t
peri=
e.. 1.
10.20 + 18. 21X + 10. 22 X + 10. 25X + 10. 24X + 1.25 X +. 10.26 X
+ et ZBX + 11. 21 1. 22 H. 23
1720
12.2
ete C
1. 24
13. 22
C2
H. 25.
12 3
e
cu
etc
30a +. 31 ax + 32aX + 33 ax + 34 a
1+ap axE + 36aX eC
ponamus dari relationem constantem inter 10, 11, 12 generalem, et similiter inter 20, 21, 2C
et intis similis inter et assumo  autem unum ampli
terminum, ut scilicet communem 10; tollam; eodem modo datur ratio constans inter 1. 21, 1122, v. 28, 1.24
dabitur et, inter 10.22 10.23, 10. 24, 10.25
et eodem modo inter 10. , 12. 20  12. 21, IL. 24, 12. 23
Ergo jam comparenti rorp +H.  n.2C eum
 + 1 1. 22 + 42. 1 =,7  1920 et 1.241 n23 +12= 34 - 1 _ 4. 20. in quibus aegus. consiteremus
considero ob telationes inter i0, 1, 1 possa tolli 12, et inter 22. 21, 20, posse tolli 2
retacta tum
i puc 23, et 24et pan sure 
i v 20 21. vdanus deinde ex his postis itaque in valoritu valore)
et hin valore 35 adeade
etc
in
en
Ergo
verte