![{\displaystyle \operatorname {Sin} .(zx,rz)\operatorname {Sin} .(xy,rx)\operatorname {Sin} .(yz,ry)=\operatorname {Sin} .(yz,rz)\operatorname {Sin} .(zx,rx)\operatorname {Sin} .(xy,ry).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/94ea868073a5e20a7a8379988ee6c3a7d3b80819)
On peut, dans l’équation (13) faire disparaître, de trois manières, deux des termes du second membre, en y supposant nulles deux des quantités
la droite
dabord de direction indéterminée, devient alors perpendiculaire à l’un des plans coordonnés, ce qui donne
![{\displaystyle r\operatorname {Cos} .(r,yz)=x\operatorname {Cos} .(x,yz),\ r\operatorname {Cos} .(r,zx)=y\operatorname {Cos} .(y,zx),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c8a141e3a9e48020fd822cbf1abe1de19e6c88e8)
![{\displaystyle r\operatorname {Cos} .(r,xy)=z\operatorname {Cos} .(z,xy).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b748cedcba50a770f1765fe976c0c0d420846a23)
(18)
En substituant les valeurs de
qui en résultent dans les équations trouvées ci-dessus, on obtiendra diverses formules indépendantes des longueurs des droites
et relatives seulement à leur direction ; les principales sont
![{\displaystyle 1={\frac {\operatorname {Cos} .(r,yz)}{\operatorname {Cos} .(x,yz}}\operatorname {Cos} .(r,x)+{\frac {\operatorname {Cos} .(r,zx)}{\operatorname {Cos} .(y,zx}}\operatorname {Cos} .(r,y)+{\frac {\operatorname {Cos} .(r,xy)}{\operatorname {Cos} .(z,xy}}\operatorname {Cos} .(r,z),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f8b5301821f1af1d42b0003fb7402ab9f8a06f6c)
![{\displaystyle {\begin{aligned}&\operatorname {Cos} .(r,x)-{\frac {\operatorname {Cos} .(r,yz)}{\operatorname {Cos} .(x,yz)}}={\frac {\operatorname {Cos} .(r,zx)}{\operatorname {Cos} .(y,zx)}}\operatorname {Cos} .(x,y)+{\frac {\operatorname {Cos} .(r,xy)}{\operatorname {Cos} .(z,xy)}}\operatorname {Cos} .(z,x),\\\\&\operatorname {Cos} .(r,y)-{\frac {\operatorname {Cos} .(r,zx)}{\operatorname {Cos} .(y,zx)}}={\frac {\operatorname {Cos} .(r,xy)}{\operatorname {Cos} .(z,xy)}}\operatorname {Cos} .(y,z)+{\frac {\operatorname {Cos} .(r,yz)}{\operatorname {Cos} .(x,yz)}}\operatorname {Cos} .(x,y),\\\\&\operatorname {Cos} .(r,z)-{\frac {\operatorname {Cos} .(r,xy)}{\operatorname {Cos} .(z,xy)}}={\frac {\operatorname {Cos} .(r,yz)}{\operatorname {Cos} .(x,yz)}}\operatorname {Cos} .(z,x)+{\frac {\operatorname {Cos} .(r,zx)}{\operatorname {Cos} .(y,zx)}}\operatorname {Cos} .(y,z).\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a1fba4a1c54e6fc03b5dd4e63c6c9a5ae2a16cf1)
![{\displaystyle 1=\left\{{\begin{aligned}&{\frac {\operatorname {Cos} .^{2}(r,yz)}{\operatorname {Cos} .^{2}(x,yz)}}+2{\frac {\operatorname {Cos} .(r,zx)\operatorname {Cos} .(r,xy)}{\operatorname {Cos} .(y,zx)\operatorname {Cos} .(z,xy)}}\operatorname {Cos} .(y,z)\\\\&{\frac {\operatorname {Cos} .^{2}(r,zx)}{\operatorname {Cos} .^{2}(y,zx)}}+2{\frac {\operatorname {Cos} .(r,xy)\operatorname {Cos} .(r,yz)}{\operatorname {Cos} .(z,xy)\operatorname {Cos} .(x,yz)}}\operatorname {Cos} .(z,x)\\\\&{\frac {\operatorname {Cos} .^{2}(r,xy)}{\operatorname {Cos} .^{2}(z,xy)}}+2{\frac {\operatorname {Cos} .(r,yz)\operatorname {Cos} .(r,zx)}{\operatorname {Cos} .(x,yz)\operatorname {Cos} .(y,zx)}}\operatorname {Cos} .(x,y)\end{aligned}}\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3b03b34a8769573f95e4c49aa2963a472d3ebb0e)
Soit encore une droite
menée, dans une direction quel-