Pagina:Werkecarlf03gausrich.djvu/67

${\tfrac {dF(u,X)}{du}}$ nancisci valorem $U',$ si statuatur $u=U,$ certo esse nequit $U'=0$. Manifesto autem $U'$ erit valor quotientis differentialis partialis ${\tfrac {dZ}{du}}$ pro $u=U$, $x=X:$ quodsi itaque insuper pro iisdem valoribus ipsarum $u,$ $x$ valorem quotientis differentialis partialis ${\tfrac {dZ}{dx}}$ per $X'$ denotemus, perspicuum est per ea quae in art. praec. demonstrata sunt, functionem $Z$ per substitutionem

$$u=U+{\tfrac {XX'}{U}}-{\tfrac {X'x}{U'}}$$

identice evanescere, adeoque per factorem

$$u+{\tfrac {X'}{U'}}x-\left(U+{\tfrac {XX'}{U'}}\right)$$

indefinite esse divisibilem. Quocirca statuendo $u=xx,$ patet, $F(xx,x)$ divisibilem esse per

$$xx+{\tfrac {X'}{U'}}x-\left(U+{\tfrac {XX'}{U'}}\right)$$

adeoque obtinere valorem $0$, si pro $x$ accipiatur radix aequationis

$$xx+{\tfrac {X'}{U'}}x-\left(U+{\tfrac {XX'}{U'}}\right)=0$$

i.e. si statuatur

$$x={\tfrac {-X\pm \surd {(4UU'U'+4XX'U'+X'X')}}{2U'}}$$

quos valores vel reales esse vel sub forma $g+h\surd {-1}$ contentos constat.

Facile iam demonstratur, per eosdem valores ipsius $x$ etiam functionem $Y$ evanescere debere. Manifesto enim $f(xx,x,\lambda ',\lambda '',\lambda '''{\text{ etc.}})$ est productum ex omnibus $(x-a)(x-b)$ exclusis repetitionibus, et proin $={\mathfrak {u}}^{m-1}.$ Hinc sponte sequitur

$${\begin{array}{c}f(xx,x,l',l'',l'''{\text{ etc.}})=y^{m-1}\\f(xx,x,L',L'',L'''{\text{ etc.}})=Y^{m-1}\end{array}}$$

sive $F(xx,x)=Y^{m-1}$, cuius itaque valor determinatus evanescere nequit, nisi simul evanescat valor ipsius $Y.$

20.

Adiumento disquisitionum praecedentium reducta est solutio aequationis $Y=0,$ i.e. inventio valoris determinati ipsius $x,$ vel realis vel sub forma $g+h\surd {-1}$ contenti, qui illi satisfaciat, ad solutionem aequationis $F(u,X)=0,$