Apart from the care of children and periods of sheer rest, the active lives were spent in the cities.
Why isn't there the in periods of sheer rest in this context?
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I think, you can consider the absence of "the" before "periods of sheer rest" being omitted. It is like when you have a list of things: a ball, pencil, desk and stone, you can use "a" before the first item and omit it before all subsequent items, because one "a" works for all.
It is not wrong to use "the" before "periods of sheer rest," but it is not necessary and it sounds repeating. Anyway, adding "the" would emphasize "periods of sheer rest."