Pagina:Werkecarlf03gausrich.djvu/146

tum sit ad $F(\alpha ^{\prime },\beta ^{\prime },\gamma ),$ denique mutando elementum tertium, donec perventum sit ad $F(\alpha ^{\prime },\beta ^{\prime },\gamma ^{\prime }),$ et perinde ab hac usque ad $F(\alpha ^{\prime \prime },\beta ^{\prime \prime },\gamma ^{\prime \prime }).$ Quum itaque per art. 7 habeantur aequationes lineares inter functionem primam, secundam atque tertiam, et generaliter inter ternas quascunque consequentes huius seriei, facile perspicitur, hinc per eliminationem deduci posse aequationem linearem inter functiones $F(\alpha ,\beta ,\gamma ),$ $F(\alpha ^{\prime },\beta ^{\prime },\gamma ^{\prime }),$ $F(\alpha ^{\prime \prime },\beta ^{\prime \prime },\gamma ^{\prime \prime }),$ ita ut generaliter loquendo e duabus functionibus, quarum tria elementa prima numeris integris differunt, quamlibet aliam functionem eadem proprietate gaudentem derivare liceat, siquidem elementum quartum idem maneat. Ceterum hic nobis sufficit, hanc veritatem insignem generaliter stabilivisse, neque hic compendiis immoramur, per quae operationes ad hunc finem necessariae quam brevissimae reddantur.

10.

Propositae sint e.g. functiones

$F(\alpha ,\beta ,\gamma ),F(\alpha +1,\beta +1,\gamma +1),F(\alpha +2,\beta +2,\gamma +2)$

inter quas aequationem linearem invenire oporteat. Iungamus ipsas per functiones contiguas sequenti modo:

${\begin{aligned}&F(\alpha ,\beta ,\gamma )=F\\&F(\alpha +1,\beta ,\gamma )=F^{\prime }\\&F(\alpha +1,\beta +1,\gamma )=F^{\prime \prime }\\&F(\alpha +1,\beta +1,\gamma +1)=F^{\prime \prime \prime }\\&F(\alpha +2,\beta +1,\gamma +1)=F^{\prime \prime \prime \prime }\\&F(\alpha +2,\beta +2,\gamma +1)=F^{\prime \prime \prime \prime \prime }\\&F(\alpha +2,\beta +2,\gamma +2)=F^{\prime \prime \prime \prime \prime \prime }\end{aligned}}$

Habemus itaque quinque aequationes lineares (e formulis 6, 13, 5 art. 7):

I.	$0=(\gamma -\alpha -1)F-(\gamma -\alpha -1-\beta )F^{\prime }-\beta (1-x)F^{\prime \prime }$
II.	$0=\gamma F^{\prime }-\gamma (1-x)F^{\prime \prime }-(\gamma -\alpha -1)xF^{\prime \prime \prime }$
III.	$0=\gamma {F}^{\prime \prime }-(\gamma -\alpha -1){F}^{\prime \prime \prime }-(\alpha +1){F}^{\prime \prime \prime \prime }$
IV.	$0=(\gamma -\alpha -1){F}^{\prime \prime \prime }-(\gamma -\alpha -2-\beta ){F}^{\prime \prime \prime \prime }-(\beta +1)(1-x){F}^{\prime \prime \prime \prime \prime }$
V.	$0=(\gamma +1){F}^{\prime \prime \prime \prime }-(\gamma +1)(1-x){F}^{\prime \prime \prime \prime \prime }-(\gamma -\alpha -1)x{F}^{\prime \prime \prime \prime \prime }$

Ex I et II prodit, eliminando $F^{\prime }$