To be perfectly specific, that’s a polka redowa, a polka step in slow waltz time.  This variation for it, called The Run, was, as far as I can tell, unique to the fifth edition (1892) of  William B. De Garmo’s The Dance of Society.

The sequence is simplicity itself:

1. In normal closed position, dance polka redowa, turning (six measures)
2. Release hands and open up into “military” position, side by side (as described and shown here).
3. Run forward six steps (two measures)
4. Join hands again to repeat from the start

De Garmo made a point of noting that the polka redowa should be danced in a gliding style as in his description of the polka step, found earlier in the book:

“…hop or spring on right foot and almost simultaneously glide left foot directly sidewise to 2d position — count one; draw right foot to 1st position — two; glide left foot again to 2d position — three…”

He also noted in the polka description that the hop “may be so modified as to be nearly imperceptible,” provided the dancers take care not to lose the elasticity of the knee.

In his description of The Run, he reiterated that it should be danced with “the three steps being glided, the glides a little longer than usual” and that the running steps should be “with spirit.”

The Run is listed as a polka redowa but the note about music reads “Mazurka Music.”  The tempo should be slower than regular waltz music, but other than that restriction, it is a flexible variation that works nicely to just about anything and in 3/4 time.

Obviously this isn’t much of a technical challenge to dance, but from a historian’s perspective, it’s interesting to have the idea that one can open up and just run forward in waltz time explicitly stated.

To vary The Run, one could use the standard nineteenth-century options of dancing forward, backward, or turning in reverse during the first six bars.  The last two, the titular “run” cannot really be altered without losing the whole concept.