Methodus inveniendi lineas curvas

272 DE CURVIS $\alpha\sqrt{(1+pp)} + \zeta p + y = -\frac{Sqq}{(1+pp)^{\frac{3}{2}}} - \sqrt{(1+pp)}\int\frac{qqdS}{(1+pp)^{\frac{5}{2}}}$ Quo fignum integrale tollamus, divisa aequatione per $\sqrt{(1+pp)}$, ea denuo differentietur; $\frac{\zeta dp}{(1+pp)^{\frac{3}{2}}} - \frac{y p dp}{(1+pp)^{\frac{3}{2}}} + \frac{2qqdS}{(1+pp)^3} + \frac{2Sqdq}{(1+pp)^2} - \frac{6Spqqdp}{(1+pp)^3} = 0$ quæ per $\frac{(1+pp)^{\frac{3}{2}}}{2q}$ multiplicata, præbet: $\frac{\zeta dp}{2q} - \frac{ypdp}{2q} + \frac{qd S + S dq}{(1+pp)^{\frac{3}{2}}} - \frac{3Spqdp}{(1+pp)^{\frac{3}{2}}} = 0;$ cujus, ob $dp=qdx$ & $dy=pdx$, integrale erit $\alpha + \frac{1}{2}\zeta x - \frac{1}{2}yy + \frac{Sq}{(1+pp)^{\frac{3}{2}}} = 0$. At eft $-\frac{(1+pp)^{\frac{3}{2}}}{q} =$ radio ofculi $R$; unde conftantes $\zeta$ & $y$ duplicando, orietur hæc aequatio $\frac{S}{R} = \alpha + \zeta x - yy$; quæ aquatio apprime congruit cum ea, quam altera Methodus directa fuppeditat. Exprimet enim $\alpha+\zeta x-yy$ momentum potentiæ incurvantis, recta quacunque pro axe affumpta, cui momento utique aequalis effe debet elafticitas abfoluta $S$ per radium ofculi $R$ divisa. Sic igitur non solum Celeb. BERNOULLII obfervata proprietas Elafticæ pleniffime eft evicta; fed etiam formularum mearum difficiliorum ufus fummus in hoc Exemplo eft declaratus. Fig. 3. 44. Si ergo curva fuerit data, quam lamina inæquabiliter elaftica a potentia $CD=P$, follicitata format; hinc elafticitas abfoluta laminæ in quovis loco poterit cognofci. Sumpta enim recta CP, quæ ad directionem vis follicitantis eft normalis, pro axe, ac pofita $CP=x$, $PM=y$, arcu curvæ $AN=s$, & radio ofculi in M $=R$, ob momentum potentiæ $P$ ad punctum M relatum $=Px$; erit $\frac{S}{R}=Px$; ideoque elafticitas abfoluta in puncto M, quæ eft $S$, $=PRx$. Hinc cum,