In Jane Eyre what do Eliza and Georgiana experience at their mother's death?

In Chapter 21 Jane returns to Gateshead to see her dying
aunt who has been asking for her. Interestingly this allows the reader ample evidence of
how Jane's character has developed in the interim since she left Gateshead all those
years ago. Despite the cold and unfriendly greeting she receives from her cousins, Jane
is not flustered. She says:

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A sneer, however, whether covert or open, had now
no longer that power over me it once
possessed...

However, to
answer your question, at the end of this Chapter, Mrs. Reed dies without either Jane or
her daughters present. At the news, Georgiana burst out weeping and said "she dared not
go" to look at her mother's corpse, so Jane and Eliza go alone. Note how Eliza
responds:

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Eliza surveyed her parent calmly. After a
silence of some minutes she observed -

"With her
constitutinon she should have lived to a good old age: her life was shortened by
trouble." And then a spasm constricted her mouth for an instant: as it passed away she
turned and left the room, and so did I. Neither of us had dropt a
tear.

Remember how Jane
summarises the excesses of her two cousins:

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Feeling without judgement is a washy draught
indeed; but judgement untempered by feeling is too bitter and husky a morsel for human
deglutition.

Georgiana, with
her obsession on getting married and emotion, represents the dangers of unfettered
feeling, whereas Eliza, with her focus on retreating from the world, represents the
extreme of judgement. Note that these two opposing forces are acted out in various
characters in the novel, though to differing extents, and most importantly, within Jane
herself. But to conclude, both sisters show their character in their response to their
mother's death - Georgiana shows an excess of feeling, and Eliza an excess of
judgement.