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Joseph Louis de Lagrange - Œuvres, Tome 5.djvu/249
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1
o
Pour les mouvements, annuels des aphélies
d
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d
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″
cos
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‴
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IV
cos
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cos
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d
φ
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d
t
=
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cos
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IV
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[
4
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′
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4
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d
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λ
′
cos
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φ
′
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φ
V
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[
5
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λ
″
cos
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″
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φ
V
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−
[
5
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λ
‴
cos
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‴
−
φ
V
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λ
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[
5
,
4
]
λ
IV
cos
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φ
IV
−
φ
V
)
λ
V
.
{\displaystyle {\begin{aligned}{\frac {d\varphi }{dt}}\ \ \ =&(0,1)+(0,2)+(0,3)+(0,4)+(0,5)\\-[0,1]&{\frac {\lambda '\cos(\varphi '-\varphi )}{\lambda }}-[0,2]{\frac {\lambda ''\cos(\varphi ''-\varphi )}{\lambda }}-[0,3]{\frac {\lambda '''\cos(\varphi '''-\varphi )}{\lambda }}\\-[0,4]&{\frac {\lambda ^{\scriptscriptstyle {\text{IV}}}\cos(\varphi ^{\scriptscriptstyle {\text{IV}}}-\varphi )}{\lambda }}-[0,5]{\frac {\lambda ^{\scriptscriptstyle {\text{V}}}\cos(\varphi ^{\scriptscriptstyle {\text{V}}}-\varphi )}{\lambda }},\\\\{\frac {d\varphi '}{dt}}\ \ =&(1,0)+(1,2)+(1,3)+(1,4)+(1,5)\\-[1,0]&{\frac {\lambda \cos(\varphi -\varphi ')}{\lambda '}}-[1,2]{\frac {\lambda ''\cos(\varphi ''-\varphi ')}{\lambda '}}-[1,3]{\frac {\lambda '''\cos(\varphi '''-\varphi ')}{\lambda '}}\\-[1,4]&{\frac {\lambda ^{\scriptscriptstyle {\text{IV}}}\cos(\varphi ^{\scriptscriptstyle {\text{IV}}}-\varphi ')}{\lambda '}}-[1,5]{\frac {\lambda ^{\scriptscriptstyle {\text{V}}}\cos(\varphi ^{\scriptscriptstyle {\text{V}}}-\varphi ')}{\lambda '}},\\\\{\frac {d\varphi ''}{dt}}\ =&(2,0)+(2,1)+(2,3)+(2,4)+(2,5)\\-[2,0]&{\frac {\lambda \cos(\varphi -\varphi '')}{\lambda ''}}-[2,1]{\frac {\lambda '\cos(\varphi '-\varphi '')}{\lambda ''}}-[2,3]{\frac {\lambda '''\cos(\varphi '''-\varphi '')}{\lambda ''}}\\-[2,4]&{\frac {\lambda ^{\scriptscriptstyle {\text{IV}}}\cos(\varphi ^{\scriptscriptstyle {\text{IV}}}-\varphi '')}{\lambda ''}}-[2,5]{\frac {\lambda ^{\scriptscriptstyle {\text{V}}}\cos(\varphi ^{\scriptscriptstyle {\text{V}}}-\varphi '')}{\lambda ''}},\\\\{\frac {d\varphi '''}{dt}}=&(3,0)+(3,1)+(3,2)+(3,4)+(3,5)\\-[3,0]&{\frac {\lambda \cos(\varphi -\varphi ''')}{\lambda '''}}-[3,1]{\frac {\lambda '\cos(\varphi '-\varphi ''')}{\lambda '''}}-[3,2]{\frac {\lambda ''\cos(\varphi ''-\varphi ''')}{\lambda '''}}\\-[3,4]&{\frac {\lambda ^{\scriptscriptstyle {\text{IV}}}\cos(\varphi ^{\scriptscriptstyle {\text{IV}}}-\varphi ''')}{\lambda '''}}-[3,5]{\frac {\lambda ^{\scriptscriptstyle {\text{V}}}\cos(\varphi ^{\scriptscriptstyle {\text{V}}}-\varphi ''')}{\lambda '''}},\\\\{\frac {d\varphi ^{\scriptscriptstyle {\text{IV}}}}{dt}}=&(4,0)+(4,1)+(4,2)+(4,3)+(4,5)\\-[4,0]&{\frac {\lambda \cos(\varphi -\varphi ^{\scriptscriptstyle {\text{IV}}})}{\lambda ^{\scriptscriptstyle {\text{IV}}}}}-[4,1]{\frac {\lambda '\cos(\varphi '-\varphi ^{\scriptscriptstyle {\text{IV}}})}{\lambda ^{\scriptscriptstyle {\text{IV}}}}}-[4,2]{\frac {\lambda ''\cos(\varphi ''-\varphi ^{\scriptscriptstyle {\text{IV}}})}{\lambda ^{\scriptscriptstyle {\text{IV}}}}}\\-[4,3]&{\frac {\lambda '''\cos(\varphi '''-\varphi ^{\scriptscriptstyle {\text{IV}}})}{\lambda ^{\scriptscriptstyle {\text{IV}}}}}-[4,5]{\frac {\lambda ^{\scriptscriptstyle {\text{V}}}\cos(\varphi ^{\scriptscriptstyle {\text{V}}}-\varphi ^{\scriptscriptstyle {\text{IV}}})}{\lambda ^{\scriptscriptstyle {\text{IV}}}}},\\\\{\frac {d\varphi ^{\scriptscriptstyle {\text{V}}}}{dt}}\ =&(5,0)+(5,1)+(5,2)+(5,3)+(5,4)\\-[5,0]&{\frac {\lambda \cos(\varphi -\varphi ^{\scriptscriptstyle {\text{V}}})}{\lambda ^{\scriptscriptstyle {\text{V}}}}}-[5,1]{\frac {\lambda '\cos(\varphi '-\varphi ^{\scriptscriptstyle {\text{V}}})}{\lambda ^{\scriptscriptstyle {\text{V}}}}}-[5,2]{\frac {\lambda ''\cos(\varphi ''-\varphi ^{\scriptscriptstyle {\text{V}}})}{\lambda ^{\scriptscriptstyle {\text{V}}}}}\\-[5,3]&{\frac {\lambda '''\cos(\varphi '''-\varphi ^{\scriptscriptstyle {\text{V}}})}{\lambda ^{\scriptscriptstyle {\text{V}}}}}-[5,4]{\frac {\lambda ^{\scriptscriptstyle {\text{IV}}}\cos(\varphi ^{\scriptscriptstyle {\text{IV}}}-\varphi ^{\scriptscriptstyle {\text{V}}})}{\lambda ^{\scriptscriptstyle {\text{V}}}}}.\end{aligned}}}
![{\displaystyle {\begin{aligned}{\frac {d\varphi }{dt}}\ \ \ =&(0,1)+(0,2)+(0,3)+(0,4)+(0,5)\\-[0,1]&{\frac {\lambda '\cos(\varphi '-\varphi )}{\lambda }}-[0,2]{\frac {\lambda ''\cos(\varphi ''-\varphi )}{\lambda }}-[0,3]{\frac {\lambda '''\cos(\varphi '''-\varphi )}{\lambda }}\\-[0,4]&{\frac {\lambda ^{\scriptscriptstyle {\text{IV}}}\cos(\varphi ^{\scriptscriptstyle {\text{IV}}}-\varphi )}{\lambda }}-[0,5]{\frac {\lambda ^{\scriptscriptstyle {\text{V}}}\cos(\varphi ^{\scriptscriptstyle {\text{V}}}-\varphi )}{\lambda }},\\\\{\frac {d\varphi '}{dt}}\ \ =&(1,0)+(1,2)+(1,3)+(1,4)+(1,5)\\-[1,0]&{\frac {\lambda \cos(\varphi -\varphi ')}{\lambda '}}-[1,2]{\frac {\lambda ''\cos(\varphi ''-\varphi ')}{\lambda '}}-[1,3]{\frac {\lambda '''\cos(\varphi '''-\varphi ')}{\lambda '}}\\-[1,4]&{\frac {\lambda ^{\scriptscriptstyle {\text{IV}}}\cos(\varphi ^{\scriptscriptstyle {\text{IV}}}-\varphi ')}{\lambda '}}-[1,5]{\frac {\lambda ^{\scriptscriptstyle {\text{V}}}\cos(\varphi ^{\scriptscriptstyle {\text{V}}}-\varphi ')}{\lambda '}},\\\\{\frac {d\varphi ''}{dt}}\ =&(2,0)+(2,1)+(2,3)+(2,4)+(2,5)\\-[2,0]&{\frac {\lambda \cos(\varphi -\varphi '')}{\lambda ''}}-[2,1]{\frac {\lambda '\cos(\varphi '-\varphi '')}{\lambda ''}}-[2,3]{\frac {\lambda '''\cos(\varphi '''-\varphi '')}{\lambda ''}}\\-[2,4]&{\frac {\lambda ^{\scriptscriptstyle {\text{IV}}}\cos(\varphi ^{\scriptscriptstyle {\text{IV}}}-\varphi '')}{\lambda ''}}-[2,5]{\frac {\lambda ^{\scriptscriptstyle {\text{V}}}\cos(\varphi ^{\scriptscriptstyle {\text{V}}}-\varphi '')}{\lambda ''}},\\\\{\frac {d\varphi '''}{dt}}=&(3,0)+(3,1)+(3,2)+(3,4)+(3,5)\\-[3,0]&{\frac {\lambda \cos(\varphi -\varphi ''')}{\lambda '''}}-[3,1]{\frac {\lambda '\cos(\varphi '-\varphi ''')}{\lambda '''}}-[3,2]{\frac {\lambda ''\cos(\varphi ''-\varphi ''')}{\lambda '''}}\\-[3,4]&{\frac {\lambda ^{\scriptscriptstyle {\text{IV}}}\cos(\varphi ^{\scriptscriptstyle {\text{IV}}}-\varphi ''')}{\lambda '''}}-[3,5]{\frac {\lambda ^{\scriptscriptstyle {\text{V}}}\cos(\varphi ^{\scriptscriptstyle {\text{V}}}-\varphi ''')}{\lambda '''}},\\\\{\frac {d\varphi ^{\scriptscriptstyle {\text{IV}}}}{dt}}=&(4,0)+(4,1)+(4,2)+(4,3)+(4,5)\\-[4,0]&{\frac {\lambda \cos(\varphi -\varphi ^{\scriptscriptstyle {\text{IV}}})}{\lambda ^{\scriptscriptstyle {\text{IV}}}}}-[4,1]{\frac {\lambda '\cos(\varphi '-\varphi ^{\scriptscriptstyle {\text{IV}}})}{\lambda ^{\scriptscriptstyle {\text{IV}}}}}-[4,2]{\frac {\lambda ''\cos(\varphi ''-\varphi ^{\scriptscriptstyle {\text{IV}}})}{\lambda ^{\scriptscriptstyle {\text{IV}}}}}\\-[4,3]&{\frac {\lambda '''\cos(\varphi '''-\varphi ^{\scriptscriptstyle {\text{IV}}})}{\lambda ^{\scriptscriptstyle {\text{IV}}}}}-[4,5]{\frac {\lambda ^{\scriptscriptstyle {\text{V}}}\cos(\varphi ^{\scriptscriptstyle {\text{V}}}-\varphi ^{\scriptscriptstyle {\text{IV}}})}{\lambda ^{\scriptscriptstyle {\text{IV}}}}},\\\\{\frac {d\varphi ^{\scriptscriptstyle {\text{V}}}}{dt}}\ =&(5,0)+(5,1)+(5,2)+(5,3)+(5,4)\\-[5,0]&{\frac {\lambda \cos(\varphi -\varphi ^{\scriptscriptstyle {\text{V}}})}{\lambda ^{\scriptscriptstyle {\text{V}}}}}-[5,1]{\frac {\lambda '\cos(\varphi '-\varphi ^{\scriptscriptstyle {\text{V}}})}{\lambda ^{\scriptscriptstyle {\text{V}}}}}-[5,2]{\frac {\lambda ''\cos(\varphi ''-\varphi ^{\scriptscriptstyle {\text{V}}})}{\lambda ^{\scriptscriptstyle {\text{V}}}}}\\-[5,3]&{\frac {\lambda '''\cos(\varphi '''-\varphi ^{\scriptscriptstyle {\text{V}}})}{\lambda ^{\scriptscriptstyle {\text{V}}}}}-[5,4]{\frac {\lambda ^{\scriptscriptstyle {\text{IV}}}\cos(\varphi ^{\scriptscriptstyle {\text{IV}}}-\varphi ^{\scriptscriptstyle {\text{V}}})}{\lambda ^{\scriptscriptstyle {\text{V}}}}}.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0db99dcb1e4c73294c8a8f60e281b73985455197)
