équations
![{\displaystyle {\begin{aligned}{\frac {r{\sqrt {p}}}{2\mathrm {K} }}\sin q=&{\sqrt {\cos(q+\beta )-\cos(q+\alpha )}},\\d\left({\frac {l{\sqrt {p}}}{\mathrm {K} }}\right)=&{\frac {dq+d\alpha }{\sqrt {\cos(q+\beta )-\cos(q+\alpha )}}}\\&\qquad \qquad -{\frac {\sin(q+\alpha )(dq+d\beta )}{\sin(q+\beta ){\sqrt {\cos(q+\beta )-\cos(q+\alpha )}}}}\\&\qquad \qquad +{\frac {r\cos q-l\cos(q+\beta )}{2\mathrm {K} \sin(q+\beta )}}{\sqrt {p}}(dq+d\beta ),\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0e5e08257397d7c352da29306d4a87d616b4160a)
![{\displaystyle {\begin{aligned}d\left({\frac {r{\sqrt {p}}}{\mathrm {K} }}\cos q\right)=&{\frac {\cos(q+\alpha )(dq+d\alpha )}{\sqrt {\cos(q+\beta )-\cos(q+\alpha )}}}\\&\qquad -{\frac {\cos(q+\beta )\sin(q+\alpha )(dq+d\beta )}{\sin(q+\beta ){\sqrt {\cos(q+\beta )-\cos(q+\alpha )}}}}\\&\qquad +{\frac {r\cos q\cos(q+\beta )-l}{2\mathrm {K} \sin(q+\beta )}}{\sqrt {p}}(dq+d\beta ).\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c992caaabd0ea18f038fc5a7f8ab41b8a00ad9ed)
§ IX.
Telles sont les équations par lesquelles on doit déterminer les forces
et
que la lame élastique
fixe en
(fig. 3, p. 86), exerce à l’extrémité
en supposant donnés la longueur
de la lame
la corde
et l’angle
Pour faciliter le calcul, on prendra la valeur de
de la première équation et on la substituera dans les deux autres, lesquelles deviendront par là
![{\displaystyle {\begin{aligned}&d\left({\frac {l{\sqrt {p}}}{\mathrm {K} }}\right)={\frac {2\mathrm {K} (dq+d\alpha )}{r{\sqrt {p}}\sin q}}\\&\quad +\left\{{\frac {2\mathrm {K} \sin(q+\alpha )}{r{\sqrt {p}}\sin q}}-{\frac {r{\sqrt {p}}\cos q}{2\mathrm {K} }}+{\frac {l{\sqrt {p}}}{2\mathrm {K} }}\left[{\frac {r^{2}p\sin ^{2}q}{4\mathrm {K} ^{2}}}+\cos(q+\alpha )\right]\right\}\\&\quad \qquad \times {\frac {8\mathrm {K} ^{2}r{\sqrt {p}}\sin qd\left(r{\sqrt {p}}\sin q\right)+16\mathrm {K} ^{4}d\cos(q+\alpha )}{16\mathrm {K} ^{4}-\left[r^{2}p\sin ^{2}q+4\mathrm {K} ^{2}\cos(q+\alpha )\right]^{2}}},\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b509e109b9cb68b5470e203d10c22ef9183c6ede)