Pagina:Werkecarlf03gausrich.djvu/51

Ita e.g. pro $m=2$ habemus

$$p=-{l'}^{2}+4{l''}$$

Perinde pro $m=3$ invenitur

$$p=-{l'}^{2}{l''}^{2}+4{l'}^{3}l'''+4{l''}^{3}-18l'l''l'''+27{l'''}^{2}$$

Determinans functionis $y$ itaque est functio coëfficientium $l',$ $l'',$ $l'''$ etc. talis, quae per substitutiones $l'=\lambda ',$ $l''=\lambda '',$ $l'''=\lambda '''$ etc. transit in productum ex omnibus differentiis inter binas quantitatum $a$, $b$, $c$ etc. In casu eo, ubi $m=1,$ i.e. ubi unica tantum indeterminata $a$ habetur, adeoque nullae omnino adsunt differentiae, ipsum numerum $1$ tamquam determinantem functionis $y$ adoptare conveniet.

In stabilienda notione determinantis, coëfficientes functionis $y$ tamquam quantitates indeterminatas spectare oportuit. Determinans functionis cum coëfficientibus determinatis

$$Y=x^{m}-L'x^{m-1}+L''x^{m-2}-L'''x^{m-3}+{\text{etc.}}$$

erit numerus determinatus $P,$ puta valor functionis $p$ pro $l'=L'$, $l''=L''$, $l'''=L'''$ etc. Quodsi itaque supponimus, $Y$ resolvi posse in factores simplices

$$Y=(x-A)(x-B)(x-C).\dots .$$

sive $Y$ oriri ex

$${\mathfrak {u}}=(x-a)(x-b)(x-c).\dots .$$

statuendo $a=A,$ $b=B,$ $c=C$ etc., adeoque per easdem substitutiones $\lambda '$, $\lambda ''$, $\lambda '''$ etc. resp. fieri $L'$, $L''$, $L'''$ etc., manifesto $P$ aequalis erit producto e factoribus

$${\begin{array}{rl}&(A-B)(A-C)(A-D).\dots .\\\times &(B-A)(B-C)(B-D).\dots .\\\times &(C-A)(C-B)(C-D).\dots .\\\times &(D-A)(D-B)(D-C).\dots .\\&{\textrm {etc.}}\end{array}}$$

Patet itaque, si fiat $P=0$, inter quantitates $A,$ $B,$ $C$ etc. duas saltem aequales reperiri debere; contra, si non fuerit $P=0,$ cunctas $A,$ $B,$ $C$ etc. necessario inaequales esse. Iam observamus, si statuamus ${\tfrac {dY}{dx}}=Y'$ sive