Page:Joseph Louis de Lagrange - Œuvres, Tome 5.djvu/242

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l’écliptique supposée fixe, et faisant $s=\theta \sin \omega ,\ u=\theta \cos \omega ,\ s'=\theta '\sin \omega ',$ $u'=\theta '\cos \omega ',\ldots ,$

{\begin{alignedat}{2}{\frac {ds}{dt}}\ \ \ &+{\Bigl [}(0,1)+(0,2)+(0,3)+(0,4)+(0,5){\Bigr ]}u\\&-(0,1)u'-(0,2)u''-(0,3)u'''-(0,4)u^{\scriptscriptstyle {\text{IV}}}-(0,5)u^{\scriptscriptstyle {\text{V}}}&=&0,\\{\frac {du}{dt}}\ \ \ &-{\Bigl [}(0,1)+(0,2)+(0,3)+(0,4)+(0,5){\Bigr ]}s\\&+(0,1)s'+(0,2)s''+(0,3)s'''+(0,4)s^{\scriptscriptstyle {\text{IV}}}+(0,5)s^{\scriptscriptstyle {\text{V}}}&=&0\,;\\{\frac {ds'}{dt}}\ \ &+{\Bigl [}(1,0)+(1,2)+(1,3)+(1,4)+(1,5){\Bigr ]}u'\\&-(1,0)u-(1,2)u''-(1,3)u'''-(1,4)u^{\scriptscriptstyle {\text{IV}}}-(1,5)u^{\scriptscriptstyle {\text{V}}}&=&0,\\{\frac {du'}{dt}}\ \ &-{\Bigl [}(1,0)+(1,2)+(1,3)+(1,4)+(1,5){\Bigr ]}s'\\&+(1,0)s+(1,2)s''+(1,3)s'''+(1,4)s^{\scriptscriptstyle {\text{IV}}}+(1,5)s^{\scriptscriptstyle {\text{V}}}&=&0\,;\\{\frac {ds''}{dt}}\ &+{\Bigl [}(2,0)+(2,1)+(2,3)+(2,4)+(2,5){\Bigr ]}u''\\&-(2,0)u-(2,1)u'-(2,3)u'''-(2,4)u^{\scriptscriptstyle {\text{IV}}}-(2,5)u^{\scriptscriptstyle {\text{V}}}&=&0,\\{\frac {du''}{dt}}\ &-{\Bigl [}(2,0)+(2,1)+(2,3)+(2,4)+(2,5){\Bigr ]}s''\\&+(2,0)s+(2,1)s'+(2,3)s'''+(2,4)s^{\scriptscriptstyle {\text{IV}}}+(2,5)s^{\scriptscriptstyle {\text{V}}}&=&0\,;\\{\frac {ds'''}{dt}}\ &+{\Bigl [}(3,0)+(3,1)+(3,2)+(3,4)+(3,5){\Bigr ]}u'''\\&-(3,0)u-(3,1)u'-(3,2)u''-(3,4)u^{\scriptscriptstyle {\text{IV}}}-(3,5)u^{\scriptscriptstyle {\text{V}}}&=&0,\\{\frac {du'''}{dt}}\ &-{\Bigl [}(3,0)+(3,1)+(3,2)+(3,4)+(3,5){\Bigr ]}s'''\\&+(3,0)s+(3,1)s'+(3,2)s''+(3,4)s^{\scriptscriptstyle {\text{IV}}}+(3,5)s^{\scriptscriptstyle {\text{V}}}&=&0\,;\\{\frac {ds^{\scriptscriptstyle {\text{IV}}}}{dt}}\ &+{\Bigl [}(4,0)+(4,1)+(4,2)+(4,3)+(4,5){\Bigr ]}u^{\scriptscriptstyle {\text{IV}}}\\&-(4,0)u-(4,1)u'-(4,2)u''-(4,3)u'''-(4,5)u^{\scriptscriptstyle {\text{V}}}&=&0,\\{\frac {du^{\scriptscriptstyle {\text{IV}}}}{dt}}\ &-{\Bigl [}(4,0)+(4,1)+(4,2)+(4,3)+(4,5){\Bigr ]}s^{\scriptscriptstyle {\text{IV}}}\\&+(4,0)s+(4,1)s'+(4,2)s''+(4,3)s'''+(4,5)s^{\scriptscriptstyle {\text{V}}}&=&0\,;\\\end{alignedat}}