Pagina:Werkecarlf03gausrich.djvu/65

quapropter $\Omega$ erit productum ex $\varphi (u,x)$ in factores

$${\begin{array}{c}1+\beta w.{\tfrac {dQ}{dx}}-\gamma w.{\tfrac {dQ}{du}}\\1+\beta 'w.{\tfrac {dQ'}{dx}}-\gamma 'w.{\tfrac {dQ'}{du}}\\1+\beta ''w.{\tfrac {dQ''}{dx}}-\gamma ''w.{\tfrac {dQ''}{du}}\end{array}}$$

etc. i.e. ex $\varphi (u,x)$ in functionem integram indeterminatarum $u,$ $x,$ $w,$ $\alpha ,$ $\beta ,$ $\gamma ,$ $\alpha ',$ $\beta ',$ $\gamma ',$ $\alpha '',$ $\beta '',$ $\gamma ''$ etc. Q.E.D.

17.

Theorema art. praec. manifesto applicabile est ad functionem $\zeta ,$ quam abhinc per

$$f(u,x,\lambda ',\lambda '',\lambda '''{\text{ etc.}})$$

exhiberi supponemus, ita ut

$$f(u+w.{\tfrac {d\zeta }{dx}},x-w.{\tfrac {d\zeta }{du}},\lambda ',\lambda '',\lambda '''{\text{ etc.}})$$

indefinite divisibilis evadat per $\zeta :$ quotientem, qui erit functio integra indeterminatarum $u,$ $x,$ $w,$ $a,$ $b,$ $c$ etc., symmetrica respectu ipsarum $a,$ $b,$ $c$ etc., exhibebimus per

$$\psi (u,x,w,\lambda ',\lambda '',\lambda '''{\text{ etc.}})$$

Hinc concludimus , fieri etiam identice

$$f(u+w.{\tfrac {dz}{dx}},x-w.{\tfrac {dz}{du}},l',l'',l'''{\text{ etc.}})=z\psi (u,x,w,l',l'',l'''{\text{ etc.}})$$

nec non

$$f(u+w.{\tfrac {dZ}{dx}},x-w.{\tfrac {dZ}{du}},L',L'',L'''{\text{ etc.}})=z\psi (u,x,w,L',L'',L'''{\text{ etc.}})$$

Quodsi itaque functionem $Z$ simpliciter exhibemus per $F(u,x),$ ita ut habeatur

$$f(u,x,L',L'',L'''{\text{ etc.}})=F(u,x)$$

erit identice

$$F\left(u+w.{\tfrac {dZ}{dx}},x-w.{\tfrac {dZ}{du}}\right)=Z\psi (u,x,w,L',L'',L'''{\text{ etc.}})$$