The ninth of a chord is, of course, not a fifth below the fifth of a chord.
However, I have just re-read the page in question from Persichetti's excellent Twentieth Century Harmony. Context is everything. This section of the book is dealing with resonance of chords, in particular in relation to the spacing of the Harmonic Series. The confusion here may be largely due to the passage being misquoted in the OP's question. Here is the full passage preceding the sentence quoted by the OP:
The principal of supporting resonance by lower sonority is occasionally applied to chordal structures. This colour device is used primarily when the composer works with chords in the upper register and needs to fill in toward the bass. In lower registers, the addition of tones is limited by the danger of muddy progressions. Most effective supporting tones are the fifth or ninth below the bottom tone of the chord because the fifth is a strong and resonant interval and the ninth is a fifth below the fifth.
Persichetti is describing notes a fifth below the existing bottom note of a chord and a ninth below the existing bottom note of a chord. This "ninth below" is a fifth below the "fifth below"! The passage below should make this clear: