Pagina:Werkecarlf03gausrich.djvu/64

Dem. Statuendo $${\begin{array}{rl}\varphi (u,x)&=(\alpha +\beta u+\gamma x)Q\\&=(\alpha '+\beta 'u+\gamma 'x)Q'\\&=(\alpha ''+\beta ''u+\gamma ''x)Q''\\&{\text{etc.}}\end{array}}$$ erunt $Q,$ $Q',$ $Q''$ etc. functiones integrae indeterminatarum $u,$ $x,$ $\alpha ,$ $\beta ,$ $\gamma ,$ $\alpha ',$ $\beta ',$ $\gamma ',$ $\alpha '',$ $\beta '',$ $\gamma '',$ etc. atque $${\begin{array}{rl}{\frac {d\varphi (u,x)}{dx}}&=\gamma Q+(\alpha +\beta u+\gamma x).{\frac {dQ}{dx}}\\&=\gamma 'Q'+(\alpha '+\beta 'u+\gamma 'x).{\frac {dQ'}{dx}}\\&=\gamma ''Q''+(\alpha ''+\beta ''u+\gamma ''x).{\frac {dQ''}{dx}}\\&{\text{etc.}}\\{\frac {d\varphi (u,x)}{du}}&=\beta Q+(\alpha +\beta u+\gamma x).{\frac {dQ}{du}}\\&=\beta 'Q'+(\alpha '+\beta 'u+\gamma 'x).{\frac {dQ'}{du}}\\&=\beta ''Q''+(\alpha ''+\beta ''u+\gamma ''x).{\frac {dQ''}{du}}\\&{\text{etc.}}\end{array}}$$ Substitutis hisce valoribus in factoribus, e quibus conflatur productum $\Omega ,$ puta in $${\begin{array}{c}\alpha +\beta u+\gamma x+\beta w.{\frac {d\varphi (u,x)}{dx}}-\gamma w.{\frac {d\varphi (u,x)}{du}}\\\alpha '+\beta 'u+\gamma 'x+\beta w.{\frac {d\varphi (u,x)}{dx}}-\gamma 'w.{\frac {d\varphi (u,x)}{du}}\\\alpha ''+\beta ''u+\gamma ''x+\beta ''w.{\frac {d\varphi (u,x)}{dx}}-\gamma ''w.{\frac {d\varphi (u,x)}{du}}\\{\text{etc. resp.}}\end{array}}$$ hi obtinent valores sequentes $${\begin{array}{c}(\alpha +\beta u+\gamma x)\left(1+\beta w.{\frac {dQ}{dx}}-\gamma w.{\frac {dQ}{du}}\right)\\(\alpha '+\beta 'u+\gamma 'x)\left(1+\beta 'w.{\frac {dQ'}{dx}}-\gamma 'w.{\frac {dQ}{du}}\right)\\(\alpha ''+\beta ''u+\gamma ''x)\left(1+\beta ''w.{\frac {dQ''}{dx}}-\gamma ''w.{\frac {dQ''}{du}}\right)\\{\text{etc.}}\end{array}}$$