Page:Joseph Louis de Lagrange - Œuvres, Tome 14.djvu/147

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CORRESPONDANCE

de

LAGRANGE  AVEC  EULER.

1.

LAGRANGE À EULER.

Taurini, 4^to cal. Julii (1754 ?)^[1].

Cogitanti mihi persæpe, ac sedulo animo inquirenti ; nunc et in differentialibus, ut in potestatibus, certus aliquis insit ordo, factum tandem est, ut in seriem a newtoniana parum discrepantem inciderim, quæ ad cujusvis gradus differentiationes æque ac integrationes possit accommodari, non secus ac illa Newtoni ad potestates, et radicalia. En itaque utrasque, primam newtonianam, alteram meam, ut si quid in ipsis inest similitudinis, totum uno oculi ictu perspiciatur :

${\begin{aligned}(a+b)^{m}=a^{m}b^{0}&+ma^{m-1}b^{1}+{\frac {m(m-1)}{2}}a^{m-2}b^{2}+{\frac {m(m-1)(m-2)}{2.3}}a^{m-3}b^{3}\\&+{\frac {m(m-1)(m-2)(m-3)}{2.3.4}}a^{m-4}b^{4}+\ldots ,\end{aligned}}$

${\begin{aligned}(xy)^{m}=x^{m}y^{0}&+mx^{m-1}y^{1}+{\frac {m(m-1)}{2}}x^{m-2}y^{2}+{\frac {m(m-1)(m-2)}{2.3}}x^{m-3}y^{3}\\&+{\frac {m(m-1)(m-2)(m-3)}{2.3.4}}x^{m-4}y^{4}+\ldots .\end{aligned}}$

Jam vero, quod ad hujusce seriei explicationem pertinet, animadver-

1. Lettres inédites, p. 5.