My initial suggestion would be that this is a "public" grave, i.e. one to which the right of burial was not owned by an individual but by the council.
Having said that, if you see nothing between 1871 and 1917, this seems an unlikely explanation as I would have thought a public grave would have been used for subsequent burials soon after the 1871 burial. Unless 37AQ is just a sub-plot within a big public plot that simply never got opened up again.
I've been trying to understand what the legal situation is with re-use in cemeteries - and that appears an unlikely explanation also! See this comment about re-use, which says "In London, local authorities have powers to reclaim a grave that has not been used for at least seventy five years, and use any space in the grave in which no interment has taken place" (my italics).
Looking at the pair of pages from the register, I believe that the explanation of what is probably (but not certainly) going on lies in the next to last column headed "If in Vault or Purchased Grave Space, No. of Grant". The 37AQ line in 1871 has no entry in that column. Therefore (setting aside the vault possibility, because @ColeValleyGirl says, from her visit, that it's a normal plot) the register indicates that this Grave Space (i.e. this plot) has not been purchased. So the council retained the right to use the plot again for another burial - which, in 1917, they exercised.
I have an inkling that in previous discussions in other places, I have seen suggestions that babies could be buried in convenient other graves. That seems to be the case here - they used a grave with room in it, to which they (the council) had the burial rights, and which had not been used for burial for 40-odd years. At least that way, the grieving parents were saved the expense of purchasing a plot.
In relation to my attempts just above to understand the legal situation with re-use - that situation does not apply, as re-use in that context must apply to grave plots where the right of burial is owned by someone other than the council concerned.
Caveat: My suggestions here are based purely on logic, not direct knowledge. This raises various questions such as
- Were 'public' graves in cemeteries ever large plots? (I had envisaged something more like a mass-grave). Or were they simply normal sized plots?
- Could someone erect a gravestone over a plot where they hadn't purchased the grave space?