A quasi-periodicity has been identified in the strange emission shifts in pulsar B1859+07 and possibly B0919+06. These events, first investigated by Rankin, Rodriguez & Wright in 2006, originally appeared disordered or random, but further mapping as well as Fourier analysis has revealed that they occur on a fairly regular basis of approximately 150 rotation periods in B1859+07 and perhaps some 700 in B0919+06. The events-which we now refer to as swooshes-are not the result of any known type of mode-changing, but rather we find that they are a uniquely different effect, produced by some mechanism other than any known pulse-modulation phenomenon. Given that we have yet to find another explanation for the swooshes, we have appealed to a last resort for periodicities in astrophysics: orbital dynamics in a binary system. Such putative companions would then have semi-major axes comparable to the light cylinder radius for both pulsars. However, in order to resist tidal disruption their densities must be at least some 10$^5$ grams/cm$^3$-therefore white-dwarf cores or something even denser might be indicated.
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