Pagina:Werkecarlf03gausrich.djvu/55

quae est functio integra indeterminatarum $x,$ $l',$ $l'',$ $l'''$ etc.

$${\text{ per }}F(x,l',l'',l'''{\text{ etc.}})$$

unde erit identice

$$1+fx.({\mathfrak {u}}-Y)+{\phi }x.{\tfrac {d({\mathfrak {u}}-Y)}{dx}}=1+F(x,\lambda ',\lambda '',\lambda '''{\text{ etc.}})$$

Habebimus itaque aequationes identicas [1]

$${\begin{array}{rl}&{\phi }a.(a-b)(a-c)(a-d)\dots .=1+F(a,\lambda ',\lambda '',\lambda '''{\text{ etc.}})\\&{\phi }b.(b-a)(b-c)(b-d)\dots .=1+F(b,\lambda ',\lambda '',\lambda '''{\text{ etc.}})\\&{\phi }c.(c-a)(c-b)(c-d)\dots .=1+F(c,\lambda ',\lambda '',\lambda '''{\text{ etc.}})\\&{\text{ etc.}}\end{array}}$$

Supponendo itaque, productum ex omnibus

$${\begin{array}{l}1+F(a,\lambda ',\lambda '',\lambda '''{\text{ etc.}})\\1+F(b,\lambda ',\lambda '',\lambda '''{\text{ etc.}})\\1+F(c,\lambda ',\lambda '',\lambda '''{\text{ etc.}})\\{\text{ etc.}}\end{array}}$$

quod erit functio integra indeterminatarum $a,$ $b,$ $c$ etc.,$l',$ $l'',$ $l'''$ etc. et quidem functio symmetrica respectu ipsarum $a,$ $b,$ $c$ etc., exhiberi per

$$\psi (\lambda ',\lambda '',\lambda '''{\text{ etc.}},l',l'',l'''{\text{ etc.}})$$

e multiplicatione cunctarum aequationum [1] resultabit aequatio identica nova [2]

$$\pi \phi a.\phi b.\phi c\dots .=\psi (\lambda ',\lambda '',\lambda '''{\text{ etc.}},\lambda ',\lambda '',\lambda '''{\text{ etc.}})$$

Porro patet, quum productum ${\phi }a.{\phi }b.{\phi }c\dots .$ indeterminatas $a,$ $b,$ $c$ etc. symmetrice involvat, inveniri posse functionem integram indeterminatarum $l',$ $l'',$ $l'''$ etc. talem, quae per substitutiones $l'=\lambda ',$ $l''=\lambda '',$ $l'''=\lambda '''$ etc. transeat in ${\phi }a.{\phi }b.{\phi }c\dots .$ Sit $t$ illa functio, eritque etiam identice [3]

$$pt=\psi (l',l'',l'''{\text{ etc.}},l',l'',l''')$$

quoniam haec aequatio per substitutiones $l'=\lambda ',$ $l''=\lambda '',$ $l'''=\lambda '''$ etc. in identicam [2] transit.

Iam ex ipsa definitione functionis $F$ sequitur, identice haberi