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Dem.
Statuendo
φ
(
u
,
x
)
=
(
α
+
β
u
+
γ
x
)
Q
=
(
α
′
+
β
′
u
+
γ
′
x
)
Q
′
=
(
α
″
+
β
″
u
+
γ
″
x
)
Q
″
etc.
{\displaystyle {\begin{array}{rl}\varphi (u,x)&=(\alpha +\beta u+\gamma x)Q\\&=(\alpha '+\beta 'u+\gamma 'x)Q'\\&=(\alpha ''+\beta ''u+\gamma ''x)Q''\\&{\text{etc.}}\end{array}}}
erunt
Q
,
{\displaystyle Q,}
Q
′
,
{\displaystyle Q',}
Q
″
{\displaystyle Q''}
etc. functiones integrae indeterminatarum
u
,
{\displaystyle u,}
x
,
{\displaystyle x,}
α
,
{\displaystyle \alpha ,}
β
,
{\displaystyle \beta ,}
γ
,
{\displaystyle \gamma ,}
α
′
,
{\displaystyle \alpha ',}
β
′
,
{\displaystyle \beta ',}
γ
′
,
{\displaystyle \gamma ',}
α
″
,
{\displaystyle \alpha '',}
β
″
,
{\displaystyle \beta '',}
γ
″
,
{\displaystyle \gamma '',}
etc. atque
d
φ
(
u
,
x
)
d
x
=
γ
Q
+
(
α
+
β
u
+
γ
x
)
.
d
Q
d
x
=
γ
′
Q
′
+
(
α
′
+
β
′
u
+
γ
′
x
)
.
d
Q
′
d
x
=
γ
″
Q
″
+
(
α
″
+
β
″
u
+
γ
″
x
)
.
d
Q
″
d
x
etc.
d
φ
(
u
,
x
)
d
u
=
β
Q
+
(
α
+
β
u
+
γ
x
)
.
d
Q
d
u
=
β
′
Q
′
+
(
α
′
+
β
′
u
+
γ
′
x
)
.
d
Q
′
d
u
=
β
″
Q
″
+
(
α
″
+
β
″
u
+
γ
″
x
)
.
d
Q
″
d
u
etc.
{\displaystyle {\begin{array}{rl}{\frac {d\varphi (u,x)}{dx}}&=\gamma Q+(\alpha +\beta u+\gamma x).{\frac {dQ}{dx}}\\&=\gamma 'Q'+(\alpha '+\beta 'u+\gamma 'x).{\frac {dQ'}{dx}}\\&=\gamma ''Q''+(\alpha ''+\beta ''u+\gamma ''x).{\frac {dQ''}{dx}}\\&{\text{etc.}}\\{\frac {d\varphi (u,x)}{du}}&=\beta Q+(\alpha +\beta u+\gamma x).{\frac {dQ}{du}}\\&=\beta 'Q'+(\alpha '+\beta 'u+\gamma 'x).{\frac {dQ'}{du}}\\&=\beta ''Q''+(\alpha ''+\beta ''u+\gamma ''x).{\frac {dQ''}{du}}\\&{\text{etc.}}\end{array}}}
Substitutis hisce valoribus in factoribus, e quibus conflatur productum
Ω
,
{\displaystyle \Omega ,}
puta in
α
+
β
u
+
γ
x
+
β
w
.
d
φ
(
u
,
x
)
d
x
−
γ
w
.
d
φ
(
u
,
x
)
d
u
α
′
+
β
′
u
+
γ
′
x
+
β
w
.
d
φ
(
u
,
x
)
d
x
−
γ
′
w
.
d
φ
(
u
,
x
)
d
u
α
″
+
β
″
u
+
γ
″
x
+
β
″
w
.
d
φ
(
u
,
x
)
d
x
−
γ
″
w
.
d
φ
(
u
,
x
)
d
u
etc. resp.
{\displaystyle {\begin{array}{c}\alpha +\beta u+\gamma x+\beta w.{\frac {d\varphi (u,x)}{dx}}-\gamma w.{\frac {d\varphi (u,x)}{du}}\\\alpha '+\beta 'u+\gamma 'x+\beta w.{\frac {d\varphi (u,x)}{dx}}-\gamma 'w.{\frac {d\varphi (u,x)}{du}}\\\alpha ''+\beta ''u+\gamma ''x+\beta ''w.{\frac {d\varphi (u,x)}{dx}}-\gamma ''w.{\frac {d\varphi (u,x)}{du}}\\{\text{etc. resp.}}\end{array}}}
hi obtinent valores sequentes
(
α
+
β
u
+
γ
x
)
(
1
+
β
w
.
d
Q
d
x
−
γ
w
.
d
Q
d
u
)
(
α
′
+
β
′
u
+
γ
′
x
)
(
1
+
β
′
w
.
d
Q
′
d
x
−
γ
′
w
.
d
Q
d
u
)
(
α
″
+
β
″
u
+
γ
″
x
)
(
1
+
β
″
w
.
d
Q
″
d
x
−
γ
″
w
.
d
Q
″
d
u
)
etc.
{\displaystyle {\begin{array}{c}(\alpha +\beta u+\gamma x)\left(1+\beta w.{\frac {dQ}{dx}}-\gamma w.{\frac {dQ}{du}}\right)\\(\alpha '+\beta 'u+\gamma 'x)\left(1+\beta 'w.{\frac {dQ'}{dx}}-\gamma 'w.{\frac {dQ}{du}}\right)\\(\alpha ''+\beta ''u+\gamma ''x)\left(1+\beta ''w.{\frac {dQ''}{dx}}-\gamma ''w.{\frac {dQ''}{du}}\right)\\{\text{etc.}}\end{array}}}