Methodus inveniendi lineas curvas

62 DE METHODO MAX. ET MIN. $— \frac{(m+1)y^{m-1}dx}{y}$: quæ multiplicata per $\frac{x^{m+1}p^n}{y^{n-1}}$ mutatur in hanc $0 = m(m+1)ydy - mnxpdy + mxydp + nx^2pdp + \frac{n(n-1)x^2p^2dy}{y} - mnxpdy$, seu $0 = m(m+1)y^2dy - 2mnxypdy + n(n-1)x^2p^2dy + mxy^2dp + nx^2ypdp$: quæ est æquatio differentialis secundi gradus, quæ, posito $y = \int vdx$ reducetur ad istam primi gradus $m(m+1)vdx + mxdu - m(2n-1)xv^2dx + nx^2vdv + n^2x^2v^2dx = 0$. Quod si autem ponamus $m=0$, ita ut maximum minimumve esse debeat $\int \frac{y^n ddy}{dy}$; habebitur hæc æquatio $(n-1)pdy+ydp=0$, quæ integrata dabit $y^{n-1}p = C$, seu $y^{n-1}dy = C [TRUNCATED]