Pagina:Werkecarlf03gausrich.djvu/63

et proin indefinite divisibilis per $\rho ,$ quum facile perspiciatur, quemlibet factorem ipsius $\rho$ in aliquo illorum factorum implicari. Statuemus itaque

$$\varphi (x,\lambda ',\lambda '',\lambda '''{\text{ etc., }}\lambda ',\lambda '',\lambda '''{\text{ etc.}})=\rho \psi (x,\lambda ',\lambda '',\lambda '''{\text{ etc.}})$$

ubi characteristica $\psi$ functionem integram exhibebit. Hinc vero facile deducitur, etiam identice esse

$$\varphi (x,L',L'',L'''{\text{ etc., }}L',L'',L'''{\text{ etc.}})=R\psi (x,L',L'',L'''{\text{ etc.}})$$

Sed supra demonstravimus, productum e factoribus

$${\begin{array}{c}f(x-a,L',L'',L'''{\text{ etc.}})\\f(x-b,L',L'',L'''{\text{ etc.}})\\f(x-c,L',L'',L'''{\text{ etc.}})\\{\text{etc.}}\end{array}}$$

quod erit $=\varphi (x,\lambda ',\lambda '',\lambda '''{\text{ etc.}},L',L'',L'''{\text{ etc.}}),$ habere terminum altissimum $N^{m}x^{m\nu };$ eundem proin terminum altissimum habebit functio $\varphi (x,L',L'',L'''{\text{ etc.}},L',L'',L'''{\text{ etc.}}),$ adeoque certo non est identice $=0.$ Quocirca etiam $R$ nequit esse identice $=0,$ neque adeo etiam determinans functionis $Z.$ Q.E.D.

16.

Theorema. Denotet $\varphi (u,x)$^[1] productum ex quotcunque factoribus talibus, in quos indeterminatae $u,$ $x$ lineariter tantum ingrediuntur, sive qui sint formae

$${\begin{array}{c}\alpha +\beta u+\gamma x\\\alpha '+\beta 'u+\gamma 'x\\\alpha ''+\beta ''u+\gamma ''x\\{\text{etc.}}\\\end{array}}$$

sit porro $w$ alia determinata. Tunc functio

$$\varphi \left(u+w.{\tfrac {d\varphi (u,x)}{dx}},x-w.{\tfrac {d\varphi (u,x)}{du}}\right)=\Omega$$

indefinite erit divisibilis per $\varphi (u,x).$

1. Vel nobis non monentibus quisque videbit, signa in art. praec. introducta restringi ad istum solum articulum, et proin significationem characterum $\varphi ,$ $w$ praesentem non esse confundendam cum pristina.