Kalkbrenner

In 1849, Friedrich Kalkbrenner published a practical instructional manual for pianists that combines the typical sequence in chord presentation with advice and examples for preluding, a nearly ubiquitous practice of improvising in the key of the composition before one performed in salon, recital, or even domestic music-making. It was only after the first World War that the practice largely died out.

Of the dominant ninth, he writes

This chord should be placed among the suspension chords, along with the eleventh and thirteenth, because it cannot be properly inverted, but because of its importance among the dissonant harmonies and the ingenious manner in which it is used, being presented in various contexts, we give it a separate chapter here. The ninth chord is nothing other than a dominant seventh chord to which a third has been added. (p. 26; my translation)

Oddly, however, Kalkbrenner's first example highlights a sequence of 9-8 suspensions, only reaching V9 at the end (boxed). Then he switches to the other extreme, with a sequence of dominant ninth chords that is very similar to one we saw in Dehn's Harmonielehre (1840).

And when he provides examples of preludes incorporating the ninth, the model (and of course the variants that follow) lack the V9 altogether:

Here are the first two variations:

The dominant ninth chord also plays no role in the nineteen exercises for preluding or the nine written-out preludes later in the volume.

Reference:

Friedrich Kalkbrenner, Traité d'harmonie du pianiste, Principes rationnels de la modulation pour apprendre à préluder et à improviser, Op. 189 (1849).  German edition: Harmonielehre zunächst für Pianofortespieler als Anleitung zum Präludieren und Improvisiren, Op. 190 (1849). Source: IMSLP; digital facsimile of a copy in the Staatsbibliothek zu Berlin.

Postscript to Dehn: Ludwig Bussler

Ludwig Bussler was a student of Siegfried Dehn, whose Harmonielehre was discussed in the previous post. Bussler was born in Berlin and spent most of his life there. He published a number of textbooks, including a Praktische Harmonielehre in Aufgaben in 1865. A posthumous fifth "revised edition" edited by Hugo Leichtentritt (1903) was available to me, along with an English translation of the second edition published in 1891.

It's hard to see any particular evidence of Dehn's influence in Bussler's dominant-ninth chapter, which is indeed practical and whose explanations and strictures are in line with most textbooks in the latter half of the 19th century.

Bussler generates the ninth chord by adding the ninth above the dominant seventh. He never mentions other extended chords. The dominant ninth is among the dissonant principal chords (Hauptakkorde), along with the dominant seventh, the diminished triad, both diminished seventh chords, and the tonic 6/4. He shows the standard resolutions, with the third (D5 in the first chord below, D4 in the second) rising to avoid parallel fifths:

He admits three inversions (the examples are from the English edition, p. 41):

but rejects the fourth "for the principle of chord-formation by successive thirds would thereby be abandoned, and the chord of the ninth be stricken out of the list of chords in a limited sense" (English trans., p. 40). He adds that the interval of the ninth must never be inverted to a seventh: "It is only necessary to strike such an inverted chord to make plain the need of this rule."

Of the following example, Bussler remarks "By the aid of this chord the descending scale can now be well harmonized between the seventh and sixth degrees" (p. 41).

Finally, he says that the ninth is best when "prepared" by appearing in the immediately preceding chord. He rejects the following (even though they were quite common in music, especially music for the stage, even at the time of his book's first edition, a point which he does not mention).

Reference:
Ludwig Bussler, Praktische Harmonielehre in Aufgaben (1865).
Ludwig Bussler, Elementary Harmony (1891), translated from the second German edition. Source: Internet Archive; digital facsimile of a copy from the University of California at Berkeley.
Ludwig Bussler, Praktische Harmonielehre in Aufgaben, Part 1 of Praktische musikalische Compositionslehre, edited by Hugo Leichtentritt (1903). Source: Internet Archive; digital facsimile of a copy from the University of Toronto. The 2-volume edition (with counterpoint as the second volume) was first published in 1878-79.

Fétis; Siegfried Dehn

By 1840, the dominant seventh chord was universally regarded as a basic harmony, but views about the ninth chord still varied. The most rigid models rejected as harmonies all dissonances other than the dominant seventh. For example,

François-Joseph Fetis “rejected the leading-tone seventh chord, all types of ninth chords, and the supertonic seventh chords. . . as true chords. Instead these verticalities, as are all other dissonant formations save for the dominant seventh, are explained using the concepts of substitution and/or retardation. Substitution is the replacement of the fifth scale degree in a dominant chord by the sixth scale degree, which produces both the dominant ninth chord and leading-tone seventh chords” (Blättler 2013, 56). 

See two examples extracted from Blättler’s Figure 1.32 below. Fétis handles inversion by expanding the common argument against the fourth inversion of the ninth chord against all inversions of the chords listed in the quote above (56-57).

Fétis was based in Brussels. During the same period, Siegfried Dehn worked in Berlin as a well-respected teacher, archivist, and editor. Dehn sought a way to include elevenths and thirteenths among the principal harmonies of a key (Hauptaccorde), a broader category than source chords (Stammaccorde), which are triads. (The dominant seventh is among the Hauptaccorde.) Dehn divides his harmony treatise in the conventional way into two parts (Capitel in the original): theoretical and practical, the first part dealing with the generation and derivation of harmonies, the second with their application to composition. The 21st of 25 chapters (§§ in the original) in Part 1 is on the ninth chord (the 22d and 23d are on the eleventh and thirteenth chords, respectively). The corresponding chapters in Part 2 are 24-26 (of 32).

As Blättler describes Dehn's method of derivation,

Ninth chords are generated . . . by placing a third both above and below the diminished triad of a key . . . ; eleventh and thirteenth chords are generated by placing the tonic note beneath the dominant seventh and ninth chords respectively. . . . These chords cannot appear in inversion nor can they sustain omissions of their roots, uppermost tones, or leading tones—restrictions [justified] on practical grounds. The extended triads cannot be inverted because their identity is dependent on a specific voicing [that is, stacks of thirds]. (p. 33)

Thus, although Dehn succeeds in placing these among the Hauptaccorde, their practical application [pace his own claims] is quite limited (Blättler, 34-35). In Part 2, chapter 24 "On the treatment of ninth chords," Dehn shows only the major and minor dominant ninths, but he does include secondary dominants. The chapter is primarily concerned with voice-leading. Here is his table of proper (regelmässige) resolutions (Dehn 1860, 215):

The important aspect of this figure is that Dehn sees no problem with a direct resolution of the major dominant ninth chord to the tonic, with its voice-leading of ^6 to ^5. This is certainly more in tune with compositional practice by 1840 than the restriction of harmonies to triads and seventh chords. He even provides a sequence with secondary dominant ninths (boxed):

Alas, when he offers a comprehensive example for all three extended chords, Dehn reverts to a chorale where those chords appear exclusively over dominant basses (boxed), effectively nullifying the ninth chord's status as an independent harmony (Dehn 1860, 219). This ambiguity and uncertainty, I note again, is typical of theoretical treatments through the century.*

* A contributing factor to the confusion is Dehn's unusual sidelining of suspensions—discussion is confined to the final chapter of Part 2, along with other non-harmonic note figures.

Reference:
Damian Blättler, "A Voicing-Centered Approach to Additive Harmony Music in France, 1889-1940," PhD dissertation, Yale University, 2013.
Siegfried Dehn, Theoretisch-praktische Harmonielehre (1840; second edition 1860). The latter was the one available to me. Its source: Digital facsimile from the Bayerische Staatsbibliothek through the Digitale Bibliothek, Münchener DigitalisierungsZentrum.
François-Joseph Fétis, Traité complet de la théorie et de la pratique de l’harmonie (1844).