Methodus inveniendi lineas curvas

ELASTICIS. 259 integratione opus erit ad laminæ incurvatæ AC longitudinem determinandam. Cum enim posito arcu $AM = s$, fit $ds = \frac{aadx}{\sqrt{(cc-xx)(2aa-cc+xx)}}$, hujus integrale, posito $x=c$, dabit longitudinem curvæ AC. 23. Cum autem istæ formulæ integrationem non admittant; per approximationem valores intervalli AD & arcus curvæ AC commode exprimere nitamur. Ponamus in hunc finem $\sqrt{(cc-xx)} = z$; eritque $PM = y = \int \frac{(aa-zz)dx}{z\sqrt{(2aa-zz)}}$, & $AM = s = \int \frac{aadx}{z\sqrt{(2aa-zz)}}$. Est vero per seriem $\frac{1}{\sqrt{(2aa-zz)}} = \frac{1}{a\sqrt{2}}(1 + \frac{1}{4} \times \frac{zz}{aa} + \frac{1.3}{4.8} \times \frac{z^4}{a^4} + \frac{1.3.5}{4.8.12} \times \frac{z^6}{a^6} + \&c.)$; unde fiet $s = \frac{1}{\sqrt{2}} \int dx (\frac{a [TRUNCATED]