Page:Annales de mathématiques pures et appliquées, 1823-1824, Tome 14.djvu/118

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{\begin{aligned}&p={\sqrt {(x-a)^{2}+(y-b)^{2}+(z-c)^{2}}},\\&q={\sqrt {(x-d)^{2}+(y-e)^{2}+(z-f)^{2}}},\\&r={\sqrt {(x-g)^{2}+(y-h)^{2}+(z-k)^{2}}},\end{aligned}}

d’où

{\begin{aligned}&\operatorname {d} p={\frac {x-a}{p}}\operatorname {d} x+{\frac {y-b}{p}}\operatorname {d} y+{\frac {z-c}{p}}\operatorname {d} z,\\\\&\operatorname {d} q={\frac {x-d}{q}}(\operatorname {d} x-\operatorname {d} d)+{\frac {y-e}{q}}(\operatorname {d} y-\operatorname {d} e)+{\frac {z-f}{q}}(\operatorname {d} z-\operatorname {d} f),\\\\&\operatorname {d} r={\frac {x-g}{r}}(\operatorname {d} x-\operatorname {d} g)+{\frac {y-h}{r}}(\operatorname {d} y-\operatorname {d} h)+{\frac {z-k}{r}}(\operatorname {d} z-\operatorname {d} k),\end{aligned}}

mais on a

{\begin{alignedat}{3}{\frac {x-a}{p}}=&\operatorname {Cos} .(p,x),\quad &{\frac {y-b}{p}}=&\operatorname {Cos} .(p,y),\quad &{\frac {z-c}{p}}=&\operatorname {Cos} .(p,z),\\\\{\frac {x-d}{q}}=&\operatorname {Cos} .(q,x),&{\frac {y-e}{q}}=&\operatorname {Cos} .(q,y),&{\frac {z-f}{q}}=&\operatorname {Cos} .(q,z),\\\\{\frac {x-g}{r}}=&\operatorname {Cos} .(r,x),&{\frac {y-h}{r}}=&\operatorname {Cos} .(r,y),&{\frac {z-k}{r}}=&\operatorname {Cos} .(r,z)\,;\end{alignedat}}

donc

{\begin{aligned}&\operatorname {d} p=\operatorname {d} x\operatorname {Cos} .(p,x)+\operatorname {d} y\operatorname {Cos} .(p,y)+\operatorname {d} z\operatorname {Cos} .(p,z),\qquad \qquad (1)\\\\&\operatorname {d} q=(\operatorname {d} x-\operatorname {d} d)\operatorname {Cos} .(q,x)+(\operatorname {d} y-\operatorname {d} e)\operatorname {Cos} .(q,y)+(\operatorname {d} z-\operatorname {d} f)\operatorname {Cos} .(q,z),\\\\&\operatorname {d} r=(\operatorname {d} x-\operatorname {d} g)\operatorname {Cos} .(r,x)+(\operatorname {d} y-\operatorname {d} h)\operatorname {Cos} .(r,y)+(\operatorname {d} z-\operatorname {d} k)\operatorname {Cos} .(r,z),\end{aligned}}

et l’on aura des valeurs analogues pour $\operatorname {d} p',\operatorname {d} q',\operatorname {d} r',\operatorname {d} p'',\operatorname {d} q'',$ $\operatorname {d} r'',\ldots$