85
PREMIÈRE PARTIE. — LIVRE I.
ces trois équations différentielles donnent les suivantes
(D)
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Ces équations sont très-commodes pour déterminer le mouvement de rotation d’un corps, lorsqu’il tourne à fort peu près autour de l’un des axes principaux, ce qui est le cas des corps célestes.
27. Les trois axes principaux auxquels nous venons de rapporter les angles
et
méritent une attention particulière ; nous allons déterminer leur position dans un solide quelconque. Les valeurs de
du numéro précédent donnent, par le n° 21, les suivantes
![{\displaystyle x''=x'\left(\operatorname {cos} \theta \operatorname {sin} \psi \operatorname {sin} \varphi +\operatorname {cos} \psi \operatorname {cos} \varphi \right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5d10b6a8af519ed1f15d6f8398f8ba6f5d7e5e97)
![{\displaystyle +y'\left(\operatorname {cos} \theta \operatorname {cos} \psi \operatorname {sin} \varphi -\operatorname {sin} \psi \operatorname {cos} \varphi \right)-z'\operatorname {sin} \theta \operatorname {sin} \varphi ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d12d167d755cd1fe08575fe4e3a639740957a730)
![{\displaystyle y''=x'\left(\operatorname {cos} \theta \operatorname {sin} \psi \operatorname {cos} \varphi -\operatorname {cos} \psi \operatorname {sin} \varphi \right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3b95d4070e5e7f1a70505c50f57cec0baf9b4c8d)
![{\displaystyle +y'\left(\operatorname {cos} \theta \operatorname {cos} \psi \operatorname {cos} \varphi +\operatorname {sin} \psi \operatorname {sin} \varphi \right)-z'\operatorname {sin} \theta \operatorname {cos} \varphi ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/42eed309311ae49e16088f0f2becb7eb963935f4)
![{\displaystyle z''=x'\operatorname {sin} \theta \operatorname {sin} \psi +y'\operatorname {sin} \theta \operatorname {cos} \psi +z'\operatorname {cos} \theta \,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/840a166f15269423bdccb8885f08de718fb2a9b6)
d’où l’on tire
![{\displaystyle {\begin{aligned}x''\operatorname {cos} \varphi -y''\operatorname {sin} \varphi &=x'\operatorname {cos} \psi -y'\operatorname {sin} \psi ,\\x''\operatorname {sin} \varphi +y''\operatorname {cos} \varphi &=x'\operatorname {cos} \theta \operatorname {sin} \psi +y'\operatorname {cos} \theta \operatorname {cos} \psi -z'\operatorname {sin} \theta .\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1d3fd25299ea56156c4c5416874c187e44eb2488)
Soit
![{\displaystyle {\begin{alignedat}{3}\mathrm {S} x'^{2}\,\ dm&=a^{2},&\qquad \mathrm {S} y'^{2}\,\ dm&=b^{2},&\qquad \mathrm {S} z'^{2}\,\ dm&=c^{2},\\\mathrm {S} x'y'dm&=f,&\mathrm {S} x'z'dm&=g,&\mathrm {S} y'z'dm&=h\,;\end{alignedat}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ccf36a2c09982e0fa7c8f701dc1bc25c46a9c2f6)
on aura
![{\displaystyle \operatorname {cos} \varphi \cdot \mathrm {S} x''z''dm-\operatorname {sin} \varphi \cdot \mathrm {S} y''z''dm}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c200687cab15544661ed2fc6a26b52e755077fc6)
![{\displaystyle =\left(a^{2}-b^{2}\right)\operatorname {sin} \theta \operatorname {sin} \psi \operatorname {cos} \psi +f\operatorname {sin} \theta \left(\operatorname {cos} ^{2}\psi -\operatorname {sin} ^{2}\psi \right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9ac59b6af16b6d44f52db5843395808ba6f4904d)
![{\displaystyle +\operatorname {cos} \theta \left(g\operatorname {cos} \psi -h\operatorname {sin} \psi \right),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/540028096fafa598546fbdc92def7c7130059f1e)
![{\displaystyle \operatorname {sin} \varphi \cdot \mathrm {S} x''z''dm+\operatorname {cos} \varphi \cdot \mathrm {S} y''z''dm}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cd3734ef1cd7224586da9946f45755514783718b)
![{\displaystyle =\operatorname {sin} \theta \operatorname {cos} \theta \left(a^{2}\operatorname {sin} ^{2}\psi +b^{2}\operatorname {cos} ^{2}\psi -c^{2}+2f\operatorname {sin} \psi \operatorname {cos} \psi \right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e42ec0bd9310af36599a908365b10d7c331929b8)
![{\displaystyle +\left({\operatorname {cos} }^{2}\,\theta -{\operatorname {sin} }^{2}\theta \right)\left(g\operatorname {sin} \psi +h\operatorname {cos} \psi \right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ccb0d6e2c326db2c0b20726471007d933e07860)