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Laplace - Œuvres complètes, Gauthier-Villars, 1878, tome 9.djvu/45
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on tirera
0
=
∂
2
z
(
1
)
∂
ϖ
∂
θ
+
m
(
1
)
∂
z
(
1
)
∂
ϖ
+
n
∂
z
(
1
)
∂
θ
+
l
(
1
)
z
(
1
)
.
{\displaystyle 0={\frac {\partial ^{2}z^{(1)}}{\partial \varpi \partial \theta }}+m^{(1)}{\frac {\partial z^{(1)}}{\partial \varpi }}+n{\frac {\partial z^{(1)}}{\partial \theta }}+l^{(1)}z^{(1)}.}
Soit
z
(
2
)
=
∂
z
(
1
)
∂
θ
+
m
(
1
)
z
(
1
)
,
{\displaystyle z^{(2)}={\frac {\partial z^{(1)}}{\partial \theta }}+m^{(1)}z^{(1)},}
et l’on aura
0
=
∂
z
(
2
)
∂
ϖ
+
n
z
(
2
)
+
z
(
1
)
(
l
(
1
)
−
∂
m
(
1
)
∂
ϖ
−
n
m
(
1
)
)
;
{\displaystyle 0={\frac {\partial z^{(2)}}{\partial \varpi }}+nz^{(2)}+z^{(1)}\left(l^{(1)}-{\frac {\partial m^{(1)}}{\partial \varpi }}-nm^{(1)}\right)\,;}
d’où, en faisant
μ
(
1
)
=
l
(
1
)
−
∂
m
(
1
)
∂
ϖ
−
n
m
(
1
)
,
m
(
2
)
=
m
(
1
)
−
∂
μ
(
1
)
∂
θ
μ
(
1
)
{\displaystyle \mu ^{(1)}=l^{(1)}-{\frac {\partial m^{(1)}}{\partial \varpi }}-nm^{(1)},\qquad m^{(2)}=m^{(1)}-{\frac {\cfrac {\partial \mu ^{(1)}}{\partial \theta }}{\mu ^{(1)}}}}
et
l
(
2
)
=
μ
(
1
)
+
n
m
(
1
)
−
n
∂
μ
(
1
)
∂
θ
μ
(
1
)
+
∂
n
∂
θ
,
{\displaystyle l^{(2)}=\mu ^{(1)}+nm^{(1)}-n{\frac {\cfrac {\partial \mu ^{(1)}}{\partial \theta }}{\mu ^{(1)}}}+{\frac {\partial n}{\partial \theta }},}
on tirera
0
=
∂
2
z
(
2
)
∂
ϖ
∂
θ
+
m
(
2
)
∂
z
(
2
)
∂
ϖ
+
n
∂
z
(
2
)
∂
θ
+
l
(
2
)
z
(
2
)
.
{\displaystyle 0={\frac {\partial ^{2}z^{(2)}}{\partial \varpi \partial \theta }}+m^{(2)}{\frac {\partial z^{(2)}}{\partial \varpi }}+n{\frac {\partial z^{(2)}}{\partial \theta }}+l^{(2)}z^{(2)}.}
Soit
z
(
3
)
=
∂
z
(
2
)
∂
θ
+
m
(
2
)
z
(
2
)
,
{\displaystyle z^{(3)}={\frac {\partial z^{(2)}}{\partial \theta }}+m^{(2)}z^{(2)},}
et l’on aura
0
=
∂
z
(
3
)
∂
ϖ
+
n
z
(
3
)
+
z
(
2
)
(
l
(
2
)
−
∂
m
(
2
)
∂
ϖ
−
n
m
(
2
)
)
;
{\displaystyle 0={\frac {\partial z^{(3)}}{\partial \varpi }}+nz^{(3)}+z^{(2)}\left(l^{(2)}-{\frac {\partial m^{(2)}}{\partial \varpi }}-nm^{(2)}\right)\,;}
d’où, en faisant
μ
(
2
)
=
l
(
2
)
−
∂
m
(
2
)
∂
ϖ
−
n
m
(
2
)
,
m
(
3
)
=
m
(
2
)
−
∂
μ
(
2
)
∂
θ
μ
(
2
)
{\displaystyle \mu ^{(2)}=l^{(2)}-{\frac {\partial m^{(2)}}{\partial \varpi }}-nm^{(2)},\qquad m^{(3)}=m^{(2)}-{\frac {\cfrac {\partial \mu ^{(2)}}{\partial \theta }}{\mu ^{(2)}}}}
et
l
(
3
)
=
μ
(
2
)
+
n
m
(
2
)
−
n
∂
μ
(
2
)
∂
θ
μ
(
2
)
+
∂
n
∂
θ
,
{\displaystyle l^{(3)}=\mu ^{(2)}+nm^{(2)}-n{\frac {\cfrac {\partial \mu ^{(2)}}{\partial \theta }}{\mu ^{(2)}}}+{\frac {\partial n}{\partial \theta }},}
on tirera
0
=
∂
2
z
(
3
)
∂
ϖ
∂
θ
+
m
(
3
)
∂
z
(
3
)
∂
ϖ
+
n
∂
z
(
3
)
∂
θ
+
l
(
3
)
z
(
3
)
.
{\displaystyle 0={\frac {\partial ^{2}z^{(3)}}{\partial \varpi \partial \theta }}+m^{(3)}{\frac {\partial z^{(3)}}{\partial \varpi }}+n{\frac {\partial z^{(3)}}{\partial \theta }}+l^{(3)}z^{(3)}.}
En continuant ainsi, on aura
0
=
∂
z
(
r
−
1
)
∂
ϖ
+
n
z
(
r
−
1
)
+
z
(
r
−
2
)
(
l
(
r
−
2
)
−
∂
m
(
r
−
2
)
∂
ϖ
−
n
m
(
r
−
2
)
)
;
{\displaystyle 0={\frac {\partial z^{(r-1)}}{\partial \varpi }}+nz^{(r-1)}+z^{(r-2)}\left(l^{(r-2)}-{\frac {\partial m^{(r-2)}}{\partial \varpi }}-nm^{(r-2)}\right)\,;}