>>Just lately, Lucy has been doing something very strange! Her tail will start to switch, she will turn and look at it, swat at it, and hiss, very emphatically! Sometimes she chases it, hissing all the while. Is she nuts? An additional piece of information is that I have another cat who recently became quite ill and is taking quite a bit of my time and attention.

This is probably not too abnormal and not a big cause for concern. Maybe Lucy is even trying to compensate for a lack of attention by playing with/by herself. If she does damage to herself or seems otherwise unhappy then you should be concerned, but if she's just hissing and playing then she's more than likely OK.
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Smile, it will make them wonder what you are up to!

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