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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folio 81r, page id 00068196:0159
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10y + Ucdy + 120V + 121dv + 100 ex  20 est aequatio quae solvi potest
(10y + iudy +120v + 121 de + 100dx =0 C Woary + mardy + B
+120 arZ
210 Y + 21 dy f220V + 221 dv L 200 dx = 0)) 120 β, 9
Sit v βy+Z, hae aequationes sic explicentur (121 dBry+i2 Brdu
et prist multiplicetur per Y, posteris per a, tL(5) 21 any + 21 aglus
et productae in unum tollantur, tollanturque y et Z 102 220 aBy
(nam ay et dZ sumil tolli non possunt)
+2200a 2
221 a d βy + 22 Bdy
+ cejaadz + 200 aadx
ifiet:
itaque fiet r= - 220a : 120, quo valore substituto in coefficiente ipsius y,
- 110. 220 a R= 120. 220 β) - 121. 220 d β 4 120. 210 a (+120. 120β) + 120. 221 aβ 20 et fiet
Habemqus ergo aequationem
dβ a, (10. 220 - 10. 210, :, 120. 221- 121. 220
in qua solae differentiae, (- 111. 220 121. 220 B + 120. 211 a + 120. 221 β) dy
(- 121. 220 + 120. 211) a4Z
(10)
(- 100. 220 + no. 200) a dx
=0
in ea
Hanc aequationem non licet reddere summabilem, nisi du et dz habeant communem
coefficientem. His iisdem positis non licet invenire aequationem in qua absit
quae fecimus
alterutra Z vel y manentibus utriusque differentialibus. nempe non licet tollere 
Redeamus ad aequationes 4 et 5, omissa multiplicatione per Y et per a, easque litaeq.oit
rursus unam, quin simul tollatur et altera
fiat 310y + 311dy + 320Z + 321 d7 + 300HX = Sudy + 5udz+ 5d
loco aequ. 5 fiat 40y + 41 dus + 420Z + 421 dqe +400 dx0
easque scribamus compendio loso aeq. 4
in his aequationibus omnes aequa tefficientes sunt quantitates datae. Eas singulas si multi-
plicemus per arbitrarias  poterimus et nempe ipsas 1, 12, 13; compositum poterimus identificare
in H9. 14 potuissemus adhuc
unam habere arbitrariam
30dy20d multiplica
per aliquam. Sed n
aequationi summabili t  710Y + 7207 +
64)
d710. y + d720. Z 7lody + 720 dZ  700a = 0
nam cum alterutra du vel dque, per alteram et dx explicari possit, facile ambae in aequatione resul-
tante ita concipiuntur, ut aequationi 14 salis fiat
1013
Cmiony tandytruoae
rulsey rustus
(15)
(6)
Qogy +quogZ+ 419dys + 421g d7Z +400gdx= 0  aequationi 14
+SIhdys +52 hdze + 500h dx 
eg 3ust ang 1 st
fit 700a= 300f + 4009. 7 500h
f1710a + d710 = 4109, : 310 L) 720 a + a d720 - 4209, : 320
unde g = a320.710 - 310.720 +320 d710 = 310d720, : 410. 320 - 310. 420 ex coincidentiatis
3
(20
et f= a, 120. 710 - 410. 720 +420 dy10 - 410 d7203 :, 310. 420 - 410. 320
C
(21
Et ex aeg. coincidentiatis du et dZ, fiet
92) , 32. 710a 31. Z2 A . 32 GAIh + 3W SuM,: 419 319-311. 421
F(23) 42. 7104 = 411. 720a - 421. 5uh + 4I1. 521h,:
2
310. 420 = 410. 320 vocetur compendio  4 10. 626
311. 42 - 4u. 321
et 311. 421 - 411. 321 rocetur compendio 100. 62
et ex 120, 22, fit +320, 621. 710 a - 310. 621.720a + 320. 621 d710 - 310. 621 d720
Et ex aequ 123 fit
= 321. 620. 710 a + 311, 620.720a
311. 521. 620 = 321, 511.620
420. 621. 710a - 410. 621. 720a + 420. 621 a d 710- 410. 621- d72
42. 620. 710a + 411. 620.720a
quos duos ipsius e valores aequando, res reducitur ad aequationem Unam, in qua
4I1. 521 62s = 421. 51I, 620
duae sunt assumtitiae 710, et 720, cum suis differentialibus, sed quia ita alterntia
pro arbitrio assumi potest, libebit unam assumere in aliqua ratione ad alterani, mu
In. 110. et ita, habetur d7o: 710 = constanti datis dato summando
si adhibuissemus in aequ. 14, aliquam maeterea arbitrariam, ut ibi annatavimus, fortasse
ller agg. 15
potuissemus efficere, ut nulla fuisset opus summatione, et potuisseit. 710 et 720 esse
poidentific
constantes. Haecarbitiaria multiplicasset retram 210 et 720 in aegus 22 et 23
si in aeq. 6 ex 4 et 3 composita volaisemus simul tollere Z et do non licuisset, sed si simul
Sidendum du in saltem, in aeg. voluissemus ullere Y et dy, ascendisemus ad est Best, B, d
quod non est in potestate. sed in altiont plurium literarum equationibus differentralibus
simplicibus credo poterimus lillere tertiam _ co et so, ita, ut jam cd equam ded, evanescat
co
qrationis
qua huc problema jam aequatio BB, (dB3, β, dB ope duarum itera differentialis sensricis duatum
interarum volvi potest vindan quae dise attior posset destrim mprsia
1ardz 4 100 arax
ta
 et Z
(25
Hol.
Imo hachon estopus