I found the death notice below while searching with Google:
"Clark County Herald", May 20, 1874
An infant of Mr. and Mrs. Clark Carothers who live in New Richmond, died last Saturday night. It was but a few day's old and was one of a pair of twin. The remaining one is not expected to live. It is so arranged by an alwise God, that the young as well as the old must be taken from us.
Little Alva Carothers lived only 10 days, but Alva's twin, Alma survived to be listed in the 1880 US Census. Beyond that, I have yet to find out what became of Alma.
An interesting piece of information is that Clark and Rachel are living in New Richmond.
Here is a useful Illinois website.