Page:Annales de mathématiques pures et appliquées, 1824-1825, Tome 15.djvu/287

${\begin{array}{l}\operatorname {Sin} .y+\operatorname {Cos} .x=+2\operatorname {Cos} .{\frac {1}{2}}\left\{{\frac {1}{2}}\varpi +(x-y)\right\}\operatorname {Cos} .{\frac {1}{2}}\left\{{\frac {1}{2}}\varpi -(x+y)\right\},(\mathrm {12} ),\\\\\operatorname {Sin} .y-\operatorname {Cos} .x=-2\operatorname {Sin} .{\frac {1}{2}}\left\{{\frac {1}{2}}\varpi +(x-y)\right\}\operatorname {Sin} .{\frac {1}{2}}\left\{{\frac {1}{2}}\varpi -(x+y)\right\},(\mathrm {13} ),\\\\\operatorname {Sin} .x+\operatorname {Cos} .y=+2\operatorname {Sin} .{\frac {1}{2}}\left\{{\frac {1}{2}}\varpi +(x-y)\right\}\operatorname {Cos} .{\frac {1}{2}}\left\{{\frac {1}{2}}\varpi -(x+y)\right\},(\mathrm {14} ),\\\\\operatorname {Sin} .x-\operatorname {Cos} .y=-2\operatorname {Cos} .{\frac {1}{2}}\left\{{\frac {1}{2}}\varpi +(x-y)\right\}\operatorname {Sin} .{\frac {1}{2}}\left\{{\frac {1}{2}}\varpi -(x+y)\right\},(\mathrm {15} ).\end{array}}$

Si, dans chacune de ces huit dernières formules, on change tour-à-tour $y$ en $y+z$ et en $y-z,$ en faisant, pour abréger,

$x+y+z=2t,\qquad$(16)

il en résultera seize autres, desquelles rejetant les quatre qui n’offrent aucune symétrie, il restera les douze suivantes :

${\begin{array}{lr}\operatorname {Sin} .x+\operatorname {Sin} .y\operatorname {Cos} .z+\operatorname {Cos} .y\operatorname {Sin} .z=+\operatorname {Sin} .t\operatorname {Cos} .(t-x),&(17)\\\\\operatorname {Cos} .x+\operatorname {Cos} .y\operatorname {Cos} .z-\operatorname {Sin} .y\operatorname {Sin} .z=+\operatorname {Cos} .t\operatorname {Cos} .(t-x),&(18)\\\\\operatorname {Sin} .x-\operatorname {Sin} .y\operatorname {Cos} .z-\operatorname {Cos} .y\operatorname {Sin} .z=-\operatorname {Cos} .t\operatorname {Sin} .(t-x),&(19)\\\\\operatorname {Cos} .x-\operatorname {Cos} .y\operatorname {Cos} .z+\operatorname {Sin} .y\operatorname {Sin} .z=+\operatorname {Sin} .t\operatorname {Sin} .(t-x),&(20)\\\\\operatorname {Cos} .x+\operatorname {Cos} .y\operatorname {Cos} .z+\operatorname {Sin} .y\operatorname {Sin} .z=+\operatorname {Cos} .(t-y)\operatorname {Cos} .(t-z),&(21)\\\\\operatorname {Cos} .x-\operatorname {Cos} .y\operatorname {Cos} .z-\operatorname {Sin} .y\operatorname {Sin} .z=-\operatorname {Sin} .(t-y)\operatorname {Sin} .(t-z),&(22)\\\\\operatorname {Sin} .x+\operatorname {Cos} .y\operatorname {Cos} .z-\operatorname {Sin} .y\operatorname {Sin} .z=\\\qquad \qquad \qquad \qquad \qquad +2\operatorname {Cos} .\left({\frac {1}{4}}\varpi -t\right)\operatorname {Sin} .\left\{{\frac {1}{4}}\varpi -(t-x)\right\},&(23)\\\\\operatorname {Cos} .x+\operatorname {Sin} .y\operatorname {Cos} .z+\operatorname {Cos} .y\operatorname {Sin} .z=\\\qquad \qquad \qquad \qquad \qquad +2\operatorname {Cos} .\left({\frac {1}{4}}\varpi -t\right)\operatorname {Cos} .\left\{{\frac {1}{4}}\varpi -(t-x)\right\},&(24)\\\\\operatorname {Sin} .x-\operatorname {Cos} .y\operatorname {Cos} .z+\operatorname {Sin} .y\operatorname {Sin} .z=\\\qquad \qquad \qquad \qquad \qquad -2\operatorname {Sin} .\left({\frac {1}{4}}\varpi -t\right)\operatorname {Cos} .\left\{{\frac {1}{4}}\varpi -(t-x)\right\},&(25)\\\\\operatorname {Cos} .x-\operatorname {Sin} .y\operatorname {Cos} .z-\operatorname {Cos} .y\operatorname {Sin} .z=\\\qquad \qquad \qquad \qquad \qquad -2\operatorname {Sin} .\left({\frac {1}{4}}\varpi -t\right)\operatorname {Sin} .\left\{{\frac {1}{4}}\varpi -(t-x)\right\},&(26)\\\\\operatorname {Sin} .x+\operatorname {Cos} .y\operatorname {Cos} .z+\operatorname {Sin} .y\operatorname {Sin} .z=\\\qquad \qquad \qquad \qquad +2\operatorname {Cos} .\left\{{\frac {1}{4}}\varpi -(t-y)\right\}\operatorname {Cos} .\left\{{\frac {1}{4}}\varpi -(t-z)\right\},&(27)\\\\\operatorname {Sin} .x-\operatorname {Cos} .y\operatorname {Cos} .z-\operatorname {Sin} .y\operatorname {Sin} .z=\\\qquad \qquad \qquad \qquad -2\operatorname {Sin} .\left\{{\frac {1}{4}}\varpi -(t-y)\right\}\operatorname {Sin} .\left\{{\frac {1}{4}}\varpi -(t-z)\right\}.&(28)\end{array}}$