Page:Annales de mathématiques pures et appliquées, 1810-1811, Tome 1.djvu/95

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LOGARITHMIQUES.

dition très-avantageuse pour la simplicité du calcul. Il faut, pour les obtenir, substituer successivement à $x$ , dans la formule U, les nombres suivans : $55,57,63,65,67,72,78,75,77,80,83,$ $85,87,88,97,125,133,153,177,245,253,287,437,525{\text{ et }}575.$

Équations.

{\begin{array}{lr}\ \ \ \mathrm {4L_{2}-\ \ L_{3}-\ \ L_{5}-\ \ L_{7}\ +2L_{11}-\ \ L_{13}-\ \ L_{29}+2L_{31}-\ \ L_{47}\cdots } &=2\mathrm {M} \left(\ \ {\tfrac {1}{37\ 20991}}+{\text{etc.}}\right)\\\ \ \ \mathrm {8L_{2}-2L_{3}+2L_{5}-2L_{7}\ -2L_{13}+2L_{19}-\ \ L_{31}\cdots } &=2\mathrm {M} \left(\ \ {\tfrac {1}{46\ 20799}}+{\text{etc.}}\right)\\\ \ \ \mathrm {2L_{2}+2L_{3}+6L_{7}\ -2L_{11}-\ \ L_{17}-\ \ L_{29}-\ \ L_{71}\cdots } &=2\mathrm {M} \left(\ \ {\tfrac {1}{84\ 70727}}+{\text{etc.}}\right)\\\ \ \ \mathrm {2L_{2}+2L_{3}-\ \ L_{7}\ +2L_{13}-\ \ L_{17}-\ \ L_{19}+2L_{29}-\ \ L_{31}-\ \ L_{73}\cdots } &=2\mathrm {M} \left({\tfrac {1}{102\ 33287}}+{\text{etc.}}\right)\\\mathrm {-5L_{2}-\ \ L_{3}-\ \ L_{5}-\ \ L_{7}\ -\ \ L_{31}+2L_{37}-\ \ L_{59}+2L_{67}\cdots } &=2\mathrm {M} \left({\tfrac {1}{122\ 90881}}+{\text{etc.}}\right)\\\mathrm {-4L_{2}+2L_{3}-\ \ L_{5}-\ \ L_{7}\ -\ \ L_{11}+2L_{13}-\ \ L_{23}-\ \ L_{67}+2L_{79}\cdots } &=2\mathrm {M} \left({\tfrac {1}{189\ 85121}}+{\text{etc.}}\right)\\\ \ \ \mathrm {4L_{2}-3L_{3}-\ \ L_{7}\ +2L_{11}-2L_{13}-\ \ L_{17}-\ \ L_{19}+2L_{73}\cdots } &=2\mathrm {M} \left({\tfrac {1}{206\ 33887}}+{\text{etc.}}\right)\\\mathrm {-3L_{2}-\ \ L_{3}+2L_{5}-\ \ L_{7}\ -\ \ L_{13}+2L_{17}+2L_{41}-\ \ L_{67}-\ \ L_{83}\cdots } &=2\mathrm {M} \left({\tfrac {1}{242\ 90449}}+{\text{etc.}}\right)\\\mathrm {-3L2-\ \ L_{3}+6L_{7}\ +2L_{11}-\ \ L_{17}-\ \ L_{23}-\ \ L_{37}-\ \ L_{41}\cdots } &=2\mathrm {M} \left({\tfrac {1}{284\ 71057}}+{\text{etc.}}\right)\\\ \ \ \mathrm {2L_{2}-\ \ L_{3}-\ \ L_{5}-\ \ L_{7}\ -2L_{11}-\ \ L_{17}+2L_{29}+2L_{73}-\ \ L_{83}\cdots } &=2\mathrm {M} \left({\tfrac {1}{358\ 53511}}+{\text{etc.}}\right)\\\mathrm {-3L_{2}+2L_{3}-\ \ L_{5}-\ \ L_{7}\ -\ \ L_{11}-2L_{13}+2L_{19}-\ \ L_{43}+2L_{83}\cdots } &=2\mathrm {M} \left({\tfrac {1}{447\ 64721}}+{\text{etc.}}\right)\\\mathrm {-3L_{2}-\ \ L_{3}-\ \ L_{7}\ -2L_{11}+2L_{13}+2L_{17}+2L_{23}-\ \ L_{31}-\ \ L_{41}\cdots } &=2\mathrm {M} \left({\tfrac {1}{516\ 73777}}+{\text{etc.}}\right)\\\ \ \ \mathrm {4L_{2}-\ \ L_{3}-\ \ L_{7}\ -\ \ L_{19}-\ \ L_{23}+2L_{29}-\ \ L_{41}+2L_{47}-\ \ L_{79}\cdots } &=2\mathrm {M} \left({\tfrac {1}{594\ 48607}}+{\text{etc.}}\right)\\\mathrm {-3L_{2}+6L_{3}-\ \ L_{7}\ +2L_{11}-\ \ L_{13}-\ \ L_{17}+2L_{19}-\ \ L_{31}-\ \ L_{83}\cdots } &=2\mathrm {M} \left({\tfrac {1}{636\ 86897}}+{\text{etc.}}\right)\\\ \ \ \mathrm {2L_{2}+2L_{3}-\ \ L_{5}-\ \ L_{7}\ +2L_{13}-\ \ L_{17}-\ \ L_{23}-\ \ L_{47}-\ \ L_{89}+2L_{97}} &=2\mathrm {M} \left({\tfrac {1}{1144\ 88711}}+{\text{etc.}}\right)\\\mathrm {-6L_{2}-\ \ L_{3}+4L_{5}-\ \ L_{7}\ +2L_{11}-2L_{13}-\ \ L_{19}+2L_{59}-\ \ L_{61}\cdots } &=2\mathrm {M} \left({\tfrac {1}{5265\ 01249}}+{\text{etc.}}\right)\\\end{array}}