Above are pictures of two consequtive measures.
It shows vi/1 - vi - V/vi - V7/iii chord progression. Doesn't it say that secondary dominant of vi, which is V/vi comes after vi, the target chord? Is secondary dominant coming after target chord also a viable analysis?
The name of the book is "The Best Music Theory for Beginners" by Dan Spencer.
This is really an abuse of Roman numeral notation. Although there are back-relating dominants, as pointed out by @Mirlan, that's not what's happening here.
Calling it V/VI- serves a practical purpose in terms of naming the chord, but not in specifying its function, which is what Roman numeral notation is designed to do. A better functional description would be IV/V/III-. That is, it's functioning as the IV chord relative to the following V7/III- chord (IV of E major = A major).
Based on the comments, I'd say it's a V7/III followed by III. In general, any major or minor chord may be preceded by its "dominant" without disturbing the local tonality. In this case, the main harmony is moving up by fifths with some chords getting a secondary dominant.
Yes, a secondary dominant can in some cases "back-relate" to its preceding tonic. A good example is Schubert's song "Mein!", where the chords (once the voice enters) go I-V7-ii-V7/ii-IV, a chromaticized version of the usual sequence I-V-ii-vi-IV.
However, I would NOT class this song in this category. Rather, iii (A minor) is preceded by its dominant, which in turn is preceded by a cadential 6/4 chord, which, unusually, is major, an instance of secondary mixture. The transition is eased by the D-minor chord, which is vi in the main tonality but a primary mixture (minor iv) in A major, hence the downward progression of the melody note F.
You example seems like a jazz/pop case, but in "classical" style you can have secondary dominants after their associated tonic chords.
In Gjerdingen's Music in the Galant Style he shows examples of various harmonic models (schemata) and their various combinations. One schema, the romanesca is based on an "opening" type harmonic gesture (I V). Another schema, the monte, is a type of harmonic sequence that moves up by steps. Both can be combines into what Gjerdingen calls the monte romanesca. One example would be analyzed as I V ii V/ii iii V/iii...
Notice in the text, Gjerdingen describes the chords as "applied [secondary] dominants of the preceding chords". So, indeed chords can be analyzed, and heard, as having dominant function coming after their target chords.
I think one of the important things to recognize with this kind of progression, with dominant after, not before, a temporary tonic, is the unresolved, or inconclusive, or "opening", or half-cadential, feeling created. You can describe it in words many ways, but it adds harmonic momentum to the music.
I'm not sure if this is what happens with your example, but the general concept is legitimate. Some other analysis of the chord in question could be better, perhaps a passing chord?
