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# Title: LBr. 56 · Bernoulli, Jac.
# Folio 65r, page id DE-611-HS-958800:0129
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Htine porro res quoque estenditur de differentias formulae triterminae, imo
et multiterminae cujuscunque. quicumque rrosintes initr  regetus homogeneorum, poterunt sic generaniter exprim:
An- t
ad eisi si duo termini unt dXdy +  dx  y. Hos multipt poterimus
Si duo ex duabus literis sint termini
quasi multiplicare per _dy pet d X ady + d’X dty, quod si rursus quasi
multiplicemus per d dX, fiet d X d’y + dX dt. Unde si terminos duos diteren-
tiales quoscunque  ex duabus literis  factos, d X d’y + d u +h. F mulipli quasi- multiplicemus
a
X. d y prodibit dX d’y + dXdty, quae est formula bitermina aa
ex similis, quam ipsi d dx d’yat day, adeoque reci differentias habens
quasi progressionis Geometrica. Cum vero terminus per quem quasi multiplicatur, nihil
mutet in differentiande, aequitur et calculo diferentiationis per quasi- potentias,
sequitur ergo et dX d’y + dx tnt, seu duas formulam bitermina biliterariam
biterminam quamcunque vi habere differentias progressionis Geometricae. Quod distinctius ita,
dX d’y + d’4  y, = quasi prod. ex, d X. d y, in dXdy + dxd y
qu
a
 ty
et
(probatur
e d, du ex d’y = de X aty = quatiu p X. py+ px p 9 in  p, p  pus + pixp y. per otersa
ductu suo
Ergo quia dX. d y (vel ipsi d sic respondens p nihil in hoc diferentiali calculo discrimnis facit
Fiet
X
dxdment quasi pxdu + Ux ammtu inv p PX+pX
Adeoque generaliter: differentiae continuata a formula bilateraria bidermina homogeneorum leges servante
c
sunt
sunt quasi progressionis Geometricae, modo hic explicato
Quosi jam smmus formulam biliterariam triterminam, itidem inter se leges lomonerectum
peniamus jam ad
servaates, sint termini a + b+c, Erit d, atb= quasi, a +b in pe aty
Et d, a+C = quosim atC, in p, xty et b+C = quasi, _ btc, in pe, X+y. Erg sed patet
esse de atb + de, atc + d btc = bis de, atbrc. Ergo fiet
bis d, atbre = quasi bra bis, a+b+c, in, p, Xty.
d a+b+C = quasi 1. atb+C, in pe, XFy
seu denique fiet
itaque seu quod idem est,
differentiae continuata formulae treferminae biliterariqe differentialis, tritermina
imo  multiterminae cujuscunque sunt quasi pro gressionis geometrilae
Et jam video me  non scabuisse opus ista tam prolixa deMonstratione
per viam gradus seundi  bitermini, ad multiterminos. praesertim
cum in ipso biter mino non sit omnibus numeris absoluta demonstratio, sed
inductione aliquousque facta progressum supponat. Hic autem ecce
demonstra tionem generatem,  pro biterminis quam multiterminis;
am
imo non tantum ut hactenus pro biliterariis, sed et pro triliterariis et
aliis quibuscunque. Nepe
Nempe d, d d d, d’xd y = quasi; p Xp y in, p, xty
Et de,  dX dy d Z = quati,,, p Xp y p Z, in, p, X+ y+ Z
Hinc x d, od X d’y dZdv, s xdey d’xda   + et =
quatibs X prys p Z dv +c d Xdey Zd V + etc, in p, XtytZt v
quaque afferentiliter expressa(intequa
Unne generaliter: differentiae formilae continuatae formula cupusque sunt progres
fit differentia
tiam ab _aggregato literarum, secundum exponentem differentiae excitatam
ex
muctliae 6 sima i
Unde patet etiam nihil turbare coefficientes. cessatus ille metus de
e coefficientibus datis non confundendis cum  numeris qui 
babu eix
a quasi potentia multiliteri simplicis oriuntur
Ex lus etiam intolligitur non esse necesse ut X et y servent leges homogenectum inter se
ab ab ab
Genecaliter ergo eo
et artium consantem interenentem a pote volli pro simsici naeficente
e
porro
C
iabega idnali
Andenique