Page:Laplace - Œuvres complètes, Gauthier-Villars, 1878, tome 3.djvu/325

${\begin{array}{cccc}&&{\rm {Exc{\grave {e}}s\ de\ ces}}&{\rm {Exc{\grave {e}}s\ des\ coefficients}}\\&&{\rm {coefficients}}&{\rm {d{\acute {e}}duits}}\\&&{\rm {sur\ ceux\ d{\acute {e}}duits\ }}&{\rm {des\ Tables\ de\ B{\ddot {u}}rg}}\\{\rm {In{\acute {e}}galit{\acute {e}}s\ d{\acute {e}}duites\quad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad }}&{\rm {Coefficients}}&{\rm {des\ Tables}}&{\rm {sur\ ceux\ d{\acute {e}}duits}}\\{\rm {des\ Tables\ de\ Mason.\quad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad }}&{\rm {de\ ma\ th{\acute {e}}orie.}}&{\rm {de\ Mason.}}&{\rm {des\ Tables\ de\ Mason.}}\end{array}}$

${\begin{alignedat}{7}&-&69''{,}&16.\sin(2gv+cv-2\theta -\varpi )^{\text{★★★}}&&-&70''{,}&86&-&\ \ 1''{,}70\quad &+&\ \ 0''{,}00\\&+&450''{,}&56.\sin(2v-2mv+cv-\varpi )^{\text{★}}&&+&453''{,}&58&+&\ \ 3''{,}02&+&\ \ 0''{,}00\\&+&44''{,}&77.\sin(2v-2mv+c'mv-\varpi ')^{\text{★}}&&+&42''{,}&02&-&\ \ 2''{,}75&+&\ \ 6''{,}17\\&-&421''{,}&37.\sin(2v-2mv-c'mv-\varpi ')^{\text{★}}&&-&415''{,}&16&+&\ \ 6''{,}21&-&\ \ 3''{,}70\\&+&67''{,}&11.\sin(2v-2mv-cv+c'mv&&+&74''{,}&96&+&\ \ 7''{,}85&-&\ \ 3''{,}40\\&&&\qquad \qquad \qquad \qquad +\varpi -\varpi ')^{\text{★}}\\&-&635''{,}&09.\sin(2v-2mv-cv-c'mv&&-&635''{,}&26&-&\ \ 0''{,}17&-&\ \ 3''{,}39\\&&&\qquad \qquad \qquad \qquad +\varpi -\varpi ')^{\text{★}}\\&+&211''{,}&84.\sin(cv+c'mv-\varpi -\varpi ')^{\text{★}}&&+&219''{,}&11&+&\ \ 7''{,}27&+&\ \ 5''{,}96\\&-&360''{,}&50.\sin(cv-c'mv-\varpi +\varpi ')^{\text{★}}&&-&362''{,}&18&-&\ \ 1''{,}68&+&\ \ 2''{,}37\\&+&551''{,}&31.\sin(2cv-2v+2mv-2\varpi )^{\text{★}}&&+&521''{,}&9i&-&29''{,}40&-&\ \ 3''{,}70\\&+&172''{,}&28.\sin(2gv-2v+2mv-2\theta )^{\text{★}}&&+&174''{,}&74&+&\ \ 2''{,}46&+&\ \ 6''{,}48\\&+&27''{,}&47.\sin(2c'mv-2\varpi ')^{\text{★}}&&+&31''{,}&25&+&\ \ 3''{,}78&-&\ \ 8''{,}95\\&+&360''{,}&12.\sin(v-mv)^{\text{★}}&&+&376''{,}&586(1&+&i)\ldots &+&17''{,}59\\&-&58''{,}&50.\sin(v-mv+c'mv-\varpi ')^{\text{★}}&&-&58''{,}&05\ \ (1&+&i)\ldots &+&16''{,}98\\\end{alignedat}}$

Inégalités du quatrième ordre et d’un ordre supérieur, qui ont été comparées aux observations.

${\begin{alignedat}{6}&-&1''{,}&03.\sin(4cv-4\varpi )^{\text{★}}&&\ldots &&\ldots &+&0''{,}09\\&-&6''{,}&10.\sin(2gv-2cv-2\theta +2\varpi )^{\text{★}}&&\ldots &&\ldots &+&0''{,}31\\&+&23''{,}&765.\sin(gv-v-\theta )^{\text{★}}&+&17''{,}135&-&\ \ 6''{,}63\quad &-&2''{,}778\\&-&21''{,}&67.\sin(3v-3mv)^{\text{★}}&&\ldots &&\ldots &+&5''{,}86\\&+&17''{,}&50.\sin(4v-4mv)^{\text{★}}&&\ldots &&\ldots &+&4''{,}63\\&+&2''{,}&38.\sin(cv+2c'mv-\varpi -2\varpi ')^{\text{★}}&&\ldots &&\ldots &-&0''{,}51\\&-&2''{,}&38.\sin(cv+2c'mv-\varpi +2\varpi ')^{\text{★}}&&\ldots &&\ldots &+&0''{,}51\\&-&27''{,}&49.\sin(2cv+2v-2mv-2\varpi )^{\text{★}}&-&\ \ 25''{,}03&+&\ \ 1''{,}66&+&2''{,}78\\&+&89''{,}&34.\sin(4v-4mv-cv+\varpi )^{\text{★}}&+&103''{,}01&+&13''{,}67&+&5''{,}55\\&+&46''{,}&79.\sin(4v-4mv-2cv+2\varpi )^{\text{★}}&+&\ \ 47''{,}71&+&\ \ 0''{,}92&-&1''{,}24\\&-&52''{,}&61.\sin(cv-v+mv-\varpi )^{\text{★}}&-&\ \ 25''{,}65.(1&+&i)\ldots &+&4''{,}01\\&-&3''{,}&53.\sin(v-mv-c'mv+\varpi ')^{\text{★}}&&\ldots &&\ldots &-&5''{,}55\\&+&29''{,}&45.\sin(2v-2mv-2gv+cv&-&\ \ 26''{,}77&-&\ \ 2''{,}68&+&1''{,}54\\&&&\qquad \qquad \qquad \qquad +2\theta -\varpi )^{\text{★}}\\&+&3''{,}&73.\sin(2gv+cv-2v+2mv&&\ldots &&\ldots &+&4''{,}94\\&&&\qquad \qquad \qquad \qquad -2\theta -\varpi )^{\text{★}}\\&-&10''{,}&87.\sin(2v-2mv-2c'mv+2\varpi ')^{\text{★★}}&&\ldots &&\ldots &-&8''{,}02\\&-&18''{,}&12.\sin(cv+v-mv-\varpi )^{\text{★★}}&-&\ \ 15''{,}47.(1&+&i)\ldots &+&8''{,}02\\&+&3''{,}&08.\sin(3cv-2v+2mv-2\varpi )^{\text{★★}}&&\ldots &&\ldots &-&6''{,}48\\&+&1''{,}&82.\sin(2v-2mv+cv+c'mv&&\ldots &&\ldots &-&6''{,}79\\&&&\qquad \qquad \qquad \qquad -\varpi -\varpi ')^{\text{★★}}\\&+&39''{,}&38.\sin(2v-2mv+cv-c'mv&+&\ \ 31''{,}39&-&\ \ 7''{,}99&-&4''{,}01\\&&&\qquad \qquad \qquad \qquad -\varpi +\varpi ')^{\text{★★}}\end{alignedat}}$