Methodus inveniendi lineas curvas

ELASTICIS. 309 quarum summa ac differentia est; $0 = A e^{\frac{a}{c}} + B \sin\frac{a}{c} - A \cos\frac{a}{c}$, seu $\frac{A}{B} = \frac{\sin\frac{a}{c}}{\cos\frac{a}{c} - e^{\frac{a}{c}}}$ $0 = B e^{-\frac{a}{c}} - A \sin\frac{a}{c} - B \cos\frac{a}{c}$, seu $\frac{A}{B} = \frac{e^{-\frac{a}{c}} - \cos\frac{a}{c}}{\sin\frac{a}{c}}$ unde fit $2 = (e^{\frac{a}{c}} + e^{-\frac{a}{c}}) \cos\frac{a}{c}$, seu $e^{\frac{a}{c}} = \frac{1 \pm \sin\frac{a}{c}}{\cos\frac{a}{c}}$ Quæ æquatio, quia congruit cum ea, quam §. 81 invenimus, sequentes Solutiones numero infinitæ satisfacient: I. $\frac{a}{c} = \frac{1}{2}\pi - \Phi = l \cot \frac{1}{2}\Phi$ II. $\frac{a}{c} = \frac{3}{2}\pi + \Phi = l \cot \frac{1}{2}\Phi$ III. $\frac{a}{c} = \frac{5}{2}\pi - \Phi = l \cot \frac{1}{2}\Phi$ &c. 96. Harum æquationum primæ satisfieri nequit, nisi fit $\Phi = 90^\circ$, ideoque $\frac{a}{c} = 0$; unde primus oscillandi modus oritur ex æquatione $\frac{a}{c} = \frac{3}{2}\pi + \Phi = l \cot \frac{1}{2}\Phi$; quæ cum jam supra sit tractata, erit $\frac{a}{c} = 4,7300350232$. Quamobrem lamina elastica, cujus uterque terminus parieti infixus tenetur, perinde vibrationes suas peraget, ac si esset omnino libera. Hæc autem Qq 3