36
AXES PRINCIPAUX
retranchant donc ce dernier résultat du précédent, on aura
![{\displaystyle AA'-B''^{2}=\int PP'(\operatorname {Cos} .\alpha \operatorname {Cos} .\beta '-\operatorname {Cos} .\alpha '\cdot \operatorname {Cos} .\beta )^{2}~;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/19db8ab1bd6f41e5419a37c9f0cd236271f7b8e6)
on aura pareillement
![{\displaystyle A'A''-B^{2}=\int PP'(\operatorname {Cos} .\beta \operatorname {Cos} .\gamma '-\operatorname {Cos} .\beta '\cdot \operatorname {Cos} .\gamma )^{2}~;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d61f598c70bd7e97a12600f7beba45c13b7f0ae3)
![{\displaystyle A''A-B'^{2}=\int PP'(\operatorname {Cos} .\gamma \operatorname {Cos} .\alpha '-\operatorname {Cos} .\gamma '\cdot \operatorname {Cos} .\alpha )^{2}~;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b3a5e91e8a9724bc9c4c6b4a116bf6d2c573a067)
donc
![{\displaystyle \int AA'-\int B^{2}=\int PP'\left\{{\begin{aligned}(\operatorname {Cos} .\alpha \cdot \operatorname {Cos} .\beta '&-\operatorname {Cos} .\alpha '\cdot \operatorname {Cos} .\beta )^{2},\\+(\operatorname {Cos} .\beta \cdot \operatorname {Cos} .\gamma '&-\operatorname {Cos} .\beta '\cdot \operatorname {Cos} .\gamma )^{2},\\+(\operatorname {Cos} .\gamma \cdot \operatorname {Cos} .\alpha '&-\operatorname {Cos} .\gamma '\cdot \operatorname {Cos} .\alpha )^{2}.\\\end{aligned}}\right.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8e1d2a844037585f4ae8f142327bd222d78035c2)
Mais, si du produit des deux premières équations
on retranche le quarré de la quatrième, on aura
![{\displaystyle (\operatorname {Cos} .\alpha \cdot \operatorname {Cos} .\beta '-\operatorname {Cos} .\alpha '\cdot \operatorname {Cos} .\beta )^{2}+(\operatorname {Cos} .\beta \cdot \operatorname {Cos} .\gamma '-\operatorname {Cos} .\beta '\cdot \operatorname {Cos} .\gamma )^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4cda90aeb05b0006830b422b8be4d1de9ef01b67)
![{\displaystyle +(\operatorname {Cos} .\gamma \cdot \operatorname {Cos} .\alpha '-\operatorname {Cos} .\gamma '\cdot \operatorname {Cos} .\alpha )^{2}=1\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/88efc36e134dfaaaea6ebbabbd081e27e871cc91)
on a donc simplement
![{\displaystyle \int AA'-\int B^{2}=\int PP',{\text{ ou }}\int PP'\int AA'-\int B^{2}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5121163cee1da150baffd88c92491fca6c6f07b3)
Il nous reste encore à trouver
; pour y parvenir formons le produit
, dans les équations
, nous aurons
![{\displaystyle AA'A''=\left\{{\begin{aligned}&\int P^{3}\operatorname {Cos} .^{2}\alpha \cdot \operatorname {Cos} .^{2}\beta \cdot \operatorname {Cos} .^{2}\gamma ,\\+&\int \mathrm {P^{2}P'} (\operatorname {Cos} .\alpha \cdot \operatorname {Cos} .^{2}\beta \cdot \operatorname {Cos} .\gamma '+\operatorname {Cos} ^{2}.\beta \cdot \operatorname {Cos} .^{2}\gamma \cdot \operatorname {Cos} .^{2}\alpha '+\operatorname {Cos} .^{2}\gamma \cdot \operatorname {Cos} .^{2}\alpha \cdot \operatorname {Cos} .\beta ')\\+&KPP'P''~;\\\end{aligned}}\right.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a1a146eeac7d9d8300c055c7062391d2ea47247)
représentant la fonction de cosinus qui multiplie ![{\displaystyle PP'P''.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a59efca175f1a846dfa7eaaa9e11fa5ab34234b1)
Effectuons aussi le produit des équations
, il viendra
![{\displaystyle BB'B''=\left\{{\begin{aligned}&\int P^{3}\operatorname {Cos} .^{2}\alpha \cdot \operatorname {Cos} .\beta \cdot \operatorname {Cos} .\gamma ,\\+&\int \mathrm {P^{2}P'} (\operatorname {Cos} .^{2}\alpha \cdot \operatorname {Cos} .\beta \cdot \operatorname {Cos} .\gamma \cdot \operatorname {Cos} .\beta '\cdot \operatorname {Cos} .\gamma '+\operatorname {Cos} .^{2}\beta \cdot \operatorname {Cos} .\gamma \cdot \operatorname {Cos} .\alpha \cdot \operatorname {Cos} .\gamma '\cdot \operatorname {Cos} .\alpha '+\ldots \\&\ldots \operatorname {Cos} .^{2}\gamma \cdot \operatorname {Cos} .\alpha \cdot \operatorname {Cos} .\beta \cdot \operatorname {Cos} .\alpha '\cdot \operatorname {Cos} .\beta ')\\+&K'PP'P''.\\\end{aligned}}\right.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/950b1139d8dce838586f18ff046a6e5c7b9fcfa7)
étant le coefficient de ![{\displaystyle PP'P''.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a59efca175f1a846dfa7eaaa9e11fa5ab34234b1)
Les équations
et
donnent encore
![{\displaystyle AB^{2}=\left\{{\begin{aligned}&\int P^{3}\operatorname {Cos} .^{2}\alpha \cdot \operatorname {Cos} .^{2}\beta \cdot \operatorname {Cos} .^{2}\gamma ,\\+&\int \mathrm {P^{2}P'} (\operatorname {Cos} .^{2}\beta \cdot \operatorname {Cos} .^{2}\gamma \cdot \operatorname {Cos} .^{2}\alpha '+2\operatorname {Cos} .^{2}\alpha \cdot \operatorname {Cos} .\beta \cdot \operatorname {Cos} .\gamma \cdot \operatorname {Cos} .\beta '\cdot \operatorname {Cos} .\gamma ')\\+&K''PP'P''~;\\\end{aligned}}\right.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dacfe4caf65d4a2d421a8bb3aeb354a2fa5bae2c)