* Throughout I am quoting from the English translation (see "reference" below).Although he introduces the system of Roman numeral labels under "keys" (Chapter 3, section Vb, §151, p. 288), Weber only begins using those labels in earnest under "modulation," one of the earliest examples being the following, one of several to demonstrate key "inertia," so that it is unlikely one would hear the fourth chord below as V7/Bb (from Chapter 4, section IIIb, §193, p. 338).
Section 1, "Of harmonic combination in general"
Section 2 "Fundamental harmonies"
Section 3, "Transformation"
Section 4, "View of the equivocalnesses arising from the transformations of the fundamental harmonies"
Section 5, "Consonances and dissonances"
Section 6, "Preparation"
Under Section 3, "Transformation":
A. Transformation of position
1. Transposition in general
2. Inversion
B. Doubling
C. Omission
D. Tones foreign to the harmony
1. Independent ninth
2. Elevation and depression of an interval
3. Transitions
4. Other tones foreign to the harmony
Ninth chords are independent when the ninth is not prepared (as it must be in suspension figures) and may possibly not be left by step, either. Here is Weber's first example of the two types of ninth chord (p. 195):
After this, he offers several examples from the contemporary repertoire (pp. 196-97):
Immediately thereafter, Weber writes that "If now, as is undeniably the fact, these examples in part have a very perceptible harshness of sound, it is, on the other hand, quite remarkable, how soon this harshness disappears, the moment the fundamental tone is omitted" (p.197; emphasis in original), as here:
This mitigation of harshness is all the explanation we ever get. Instead, Weber goes on for several pages on various ways to distinguish the ninth without its root from the seventh chord that looks just like it. (In keeping with a central tenet of his theory, ambiguity—or "equivocalness," as in one of the headings above—he says at one point that sometimes you can't tell them apart.)
Reference:
Gottfried Weber, Versuch einer geordneten Theorie der Tonsetzkunst (1817-21). English translation of the 3d edition by James F. Warner as An Attempt at a Systematically Arranged Theory of Musical Composition (2d ed, 1842). Source: IMSLP; digital facsimile of a copy in the Yale University Library.