John Bedington (ex Charlecote Watermill) tells me that he has only once seen this arrangement, on the brake wheel and wallower in the windmill now at Madingley, Cambridgeshire (removed from Ellington, Huntingdonshire).  He also mentioned that Rex Wailes, in “The English Windmill” seems to treat this example as a one-off for windmills and says ‘the brake wheel and wallower have two rows of staggered cogs, evidently to avoid backlash’.

The double row of cogs would be stronger than one row, but the point of staggering the teeth would seem to be to reduce backlash in the gears (that is, rattle, because of play between the cogs).  The double row would reduce noise and possibly make the cogs less likely to break.  It would also have the same effect as having a finer pitch of cog.

I suspect other examples may exist;  do let me know if you know of any.  nigel.harrismsc@gmail.com

Editor’s note: 

Keith says that twin staggered teeth are a feature of the Abt system of rack railways.  It is used on the Snowdon Mountain Railway and the Gornergrat railway at Zermatt (illustrated), for example.  As well as the advantages that you mention, it also means that the gear wheel on the locomotive and the rack (and in the case of the mill, the two gear wheels) are in more constant mesh, and therefore will be safer because they are less likely to jump out of engagement.  (This is possibly more important in a rack railway than a mill.)