Page:Annales de mathématiques pures et appliquées, 1814-1815, Tome 5.djvu/348

340

DE LA LIGNE DROITE ET DU PLAN.

${\begin{aligned}&\left.{\begin{aligned}\Pi \operatorname {Sin} .(y,z)\operatorname {Cos} .(pq,yz)=A\operatorname {Sin} .^{2}(y,z)&-B\left\{\operatorname {Cos} .(z,x)-\operatorname {Cos} .(x,y)\operatorname {Cos} .(y,z)\right\}\\&-C\left\{\operatorname {Cos} .(x,y)-\operatorname {Cos} .(y,z)\operatorname {Cos} .(z,x)\right\}\\\end{aligned}}\right\},\\&\left.{\begin{aligned}\Pi \operatorname {Sin} .(z,x)\operatorname {Cos} .(pq,zx)=B\operatorname {Sin} .^{2}(z,x)&-C\left\{\operatorname {Cos} .(x,y)-\operatorname {Cos} .(y,z)\operatorname {Cos} .(z,x)\right\}\\&-A\left\{\operatorname {Cos} .(y,z)-\operatorname {Cos} .(z,x)\operatorname {Cos} .(x,y)\right\}\\\end{aligned}}\right\},\\&\left.{\begin{aligned}\Pi \operatorname {Sin} .(x,y)\operatorname {Cos} .(pq,xy)=C\operatorname {Sin} .^{2}(x,y)&-A\left\{\operatorname {Cos} .(y,z)-\operatorname {Cos} .(z,x)\operatorname {Cos} .(x,y)\right\}\\&-B\left\{\operatorname {Cos} .(z,x)-\operatorname {Cos} .(x,y)\operatorname {Cos} .(y,z)\right\}\\\end{aligned}}\right\}.\\\end{aligned}}$

au moyen de quoi la formule (23) deviendra

$\operatorname {Cos} .(pq,p'q')={\frac {1}{\Pi '}}\left\{{\begin{aligned}&A'\operatorname {Sin} .(y,z)\operatorname {Cos} .(pq,yz)\\+&B'\operatorname {Sin} .(z,x)\operatorname {Cos} .(pq,zx)\\+&C'\operatorname {Sin} .(x,y)\operatorname {Cos} .(pq,xy)\\\end{aligned}}\right\}\,;\qquad$(24)

ou encore

$\operatorname {Cos} .(pq,p'q')={\frac {1}{\Pi }}\left\{{\begin{aligned}&A\operatorname {Sin} .(y,z)\operatorname {Cos} .(yz,p'q')\\+&B\operatorname {Sin} .(z,x)\operatorname {Cos} .(zx,p'q')\\+&C\operatorname {Sin} .(x,y)\operatorname {Cos} .(xy,p'q')\\\end{aligned}}\right\}\,;\qquad$(25)

mais, en éliminant $\Delta$ entre les formules (12) et (14), il vient

${\begin{aligned}{\frac {A\operatorname {Sin} .(y,z)}{\Pi }}=&{\frac {\operatorname {Sin} .(pq,x)}{\operatorname {Sin} .(yz,x)}},\\{\frac {B\operatorname {Sin} .(z,x)}{\Pi }}=&{\frac {\operatorname {Sin} .(pq,y)}{\operatorname {Sin} .(zx,y)}},\\{\frac {C\operatorname {Sin} .(x,y)}{\Pi }}=&{\frac {\operatorname {Sin} .(pq,z)}{\operatorname {Sin} .(xy,z)}}\,;\\\end{aligned}}$

substituant ces valeurs dans la forme (25), elle deviendra

$\operatorname {Cos} .(pq,p'q')={\frac {\operatorname {Sin} .(pq,x)}{\operatorname {Sin} .(yz,x)}}\operatorname {Cos} .(yz,p'q')+{\frac {\operatorname {Sin} .(pq,y)}{\operatorname {Sin} .(zx,y)}}\operatorname {Cos} .(zx,p'q')$

$+{\frac {\operatorname {Sin} .(pq,z)}{\operatorname {Sin} .(xy,z)}}\operatorname {Cos} .(xy,p'q').\qquad$(26)

Si, au moyen des conditions (II), on fait successivement coïncider les deux plans $pq,p'q'$ avec deux plans coordonnés différens, on tirera des formules (23)