Polarization: etymology question

The polarization identity expresses a symmetric bilinear form on a vector space in terms of its associated quadratic form: $$ \langle v,w\rangle = \frac{1}{2}(Q(v+w) - Q(v) - Q(w)), $$ where $Q(v) = \langle v,v\rangle$. More generally (over fields of characteristic $0$), for any homogeneous polynomial $h(x_1,\dots,x_n)$ of degree $d$ in $n$ variables, there is a unique symmetric $d$-multilinear polynomial $F({\mathbf x}_1,\dots,{\mathbf x}_d)$, where each ${\mathbf x}_i$ consists of $n$ indeterminates, such that $h(x_1,\dots,x_n) = F({\mathbf x},\dots,{\mathbf x})$, where ${\mathbf x} = (x_1,\dots,x_n)$. There is a formula which expresses $F({\mathbf x}_1,\dots,{\mathbf x}_d)$ in terms of $h$, generalizing the above formula for a bilinear form in terms of a quadratic form, and it is also called a polarization identity.

Where did the meaning of "polarization", in this context, come from? Weyl uses it in his book The classical groups (see pp. 5 and 6 on Google books) but I don't know if this is the first place it appeared. Jeff Miller's extensive math etymology website doesn't include this term. See http://jeff560.tripod.com/p.html.

Where did the meaning of "polarization", in this context, come from? Weyl uses it in his book The classical groups (see pp. 5 and 6 on Google books) but I don't know if this is the first place it appeared.

A few things I've managed to find ...

The term polarization in this context did not originate with Weyl (1939). The book Hilbert's Invariant Theory Papers is an English translation of four papers by David Hilbert, and the term "polarization" appears in the first two of them (published 1885 and 1887), evidently in the sense you have in mind. In the fourth paper (published 1893), Hilbert uses an expression that translates as "Aronhold process" for what Hawkins' Emergence of the theory of Lie groups: an essay in the history of mathematics, 1869-1926 terms the "Aronhold polarization process". Also, Gordan (1885) refers to this same "Aronhold process", which was apparently published in 1838 (if not also earlier) by Aronhold.

In the above-cited works, the meaning of polarization appears to derive from that of the terms pole and polar as used in projective geometry. The entry for "POLE and POLAR" on the webpage by Jeff Miller, mentioned in the question, says the term pôle in this sense was introduced by François Joseph Servois in 1811, and that the term polar (polaire) was introduced by Joseph-Diez Gergonne in the modern geometric sense in 1813.