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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folio 88v, page id 00068196:0174
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Esto aequatio data dy: x = XX + gy,: aa,  seu
(2)
i3 qaadys +v agdy =d
vi 12, 10, 11 suyt
fet ly :dx= 22aa + Elayt 20gy,:, tiaa F quay ~ 2XX +9y,: aa, et habebimus aequationem
tr. 22a F, 2103y +u2oaayy
20
in auxx+ qa xy 4 aayys + 28 ay3
vidermus an hic utiliter liceat facere 4) = 11. mar. X1 7X
etfiei 12, 1. 21 a = 11. 20ay + 10.40 yy 120
10. H ay
dA. D + ac du +dH 20 aydx + 102gt  ydx + 1. 2onady + 10. qu.ydy 0
Hanc aeq. 7. differentiando, siet:
U d21 aa do 11 d2o dy dx
2.1X(12 20 dx A. Et ay ex
12Xx dqudx 10 dal ay ax
ut (80
10 d4o gylx 10 4 adus
(ullistinti uite e
eu rindeltiglicata per 60 addita aeg. 2ib
Edinoutst 1  nabru 4 100 obyaue 2240
00. 6oyydx -00.60 aadu
fier 40F0 sed ita etiam frertet 6020
quid qun biecto itaque vereor, ut liceat tollere temminum absolutum in aeu5. Eo ergo retento
00. 60 XXdx +
duc  du. 21 cayax L dureang X tdlo. 40 ydx + 11. 21 a dy 7 H. 20n ay dy , 10. 40 agy dis
si differentiamus fiet
pde Ta du
d
Bx
 Udro
12 d.. B2XX 140
a10 4
101a
10d4d
12 4 eb 2. 10. 41 aaydy
140. axedy
comparandum crum aequ. 1. multii l. per 609+61a
bi axplx + bxpydx + biagydu + 60 2 dx -6qa dus - 6causny ce
b=0
iat
fet b2 2. 10 =29 et ex dy fit  2.  1941, 11. 20 = 60, seu 41- 20 4 60
ex joste
 Ex gydx fit d40 = (2) 20 + 41 (-20) Ex y, d28 7 d 4) 1)2. 21
d11 =0
1
denius ex ex sit d21 + 2X 2. 40X =+ X d qu 2.20 + 4, - 20, antitti potest aequatio hac
ultrum qoia sola continet 22 item ag. prim ex is 12, quia sola continet d4
Porro ob sublatum terminum altonum in aeg. 5 Z2= AIXX: aa Ergo ex aeg. 1+ faet
d21. F 2. 40X + XX d40= - 2. axX + 41 - 20 Res ergo redit ad, ce gy. 12. 13, 16
seu 2072521 - 4
d2L ex aeq. 16, d21 + 4sset 4. X() 21 + XX2(2 - - 2. 4 1XX  + 41 - 2 s21 + 41 - si Aad per ta set +. EF sat = L XF L a1X + a - 2( it + 41, et differtulando fer.
ponamus 21 esse cognitam pro arbirio assumendam Ex 13 fiet 20+ 41= 2521 Ergo
res redit ad aeqqu. 12) 16, 17. Eit ex 12 fiet d40 = 2 sel Et 40 = 2 s521. superest
et habeber I, atque adeo cum 21 pro arbitio assumi possit habetur quaesitum sea quaesita 4I
la
potest poni 21 constans adeoque (21 = 21x et B2 =  2. U XX
Assumitur alia differentialis ANX A- 4 1xax + 2 a ydx + 20 y ux + N aatus + 40 a y dy = (2) = 0. Et fiet
ex 1; - d g: dx = XXF yy, : - aa = (per 2) - MIXX + 21ay + 20yy, :, 4aa + 4ugy unde prodit aequatio
+40xxA + 41 ay + 40 yy = (4) =0
F 41 ad XX) + 4 cl Xxy + 41 A yy + 40y3. 720 vel +ila  + 20 ad
= AI a XX )+ 21 aay + 20 a gy
unde differentiqudo fit  2 4 Xdx + d H ay. + d to yy rodau
ago XXx +d 20 ayx
Haec aequatio coincidetur datae, multiplicatae perbo,
cui addatur assumta 2, multi plicata per b, sart
arlaaix
60 xxdx
.
+ 60 gydx -b aadus
7 = (9 =0
et fiet .. ...
∞ AIb xxdx +2by dx + 20b yyax +alabays +40 b ydy s
Iustituta jam cincidentiatione t ob ydu, sit b = 2. Et ob du fit (2)+ 20 = - 60 (+2) 41 Erit 60 = 4 2(7) = 41 - 20
vyarex
ob yydx fit 440 = 60 +2. 20 seu dgo = 41 - 20 +2. 20 seu d90 = 4-  (8) 41 + 20 bC f
Ax
ob ydx fit d4l + d20 = 2 t (9) = 2. 21 denique ob dxx fit 2. +oX +  ex d G0XX + d2 I = 1) ( 1 - 2C) XX. s
redit ad solvendas aequationes 8, 9, 10. Sit 2 quantitas constans, seu d21 = (11) = 0, f. eth
Et ex 9 fier 41 +20 = K (12) = 2(21. Ergo Ex 8 fit 440  seu A 25i1 et 40 (14) = 2s21. quibus
valoribus ex 11, 13, 14 substitutis in aeg. 10, fiet 2421 + 2XX21 =  = 41∞ 2OXX - 2. 4 1XX)   -20xp - 21xx
andesit 2X (21 + 2XX) 21 - - 2XX s21 =0 seu B21 + 2Xs21 = d seu (s21) = (2) 521 + 2 1X seu 21+ 2 21 dx + 2X d20
Ad qua 221 =0, hoc non licet, hisi sit 21 = 0. Hoc pacto ex aeq.q fiet d41 +d20 = (17) =0 Et 41 +20 = (18) =c
Ergo et d40 9) = C et 40 = (20) = CX sed ita ex aeq. 10 fiet 2E XX + CXX = (- 41+ 20) = Z Cxx, quod
est atsurdum Ergo nonlicet facere 21 constantem nempe 41 et 20, non constituunt nisi unam quantitatem 21 + 20
itaque in aeq. 10 fiet (2 + 2521 + d21 = - X∞ C 2Xs21. seu 2X (s21 + aex se1 + d210 seu
25
et posito 131.  fiet s21 = dg et 21 = ddgu, et d 21 = d3’g, et fiit 29X + 4XX d + d38 = 0.
si esset d3g d’ =0 fimet 2X = - 2d8 seu log X = logi
ex est conlans non potest ’esse constans, C non litel faicere 2ca = - 41XX ad deprimend a, ey.C
et  d = dv et d(() = b lv. et ddlo - b dv acv b
A 8- V Bo sia d log  f ag L hg bts v et a V- vB  s   =  do b + dtotr b + de dso. lul epupuirbid qu nus sen foe tile prio sim letepes ater