David,
Your situation is handled through a child table. The first normal form states that it is achieved by removing repeating or multivalued attributes to a child table.
In your scenario the Mother is one attribute and the others are children of the mother. You can see that every other field in teh table design is a child of the mother (they all have the same domain definition and therefore are the same attribute - a multivalued attribute).
To normalize this table you would create a table for the children that is related to the mother table and allow one record in the children table for each child of a mother.
Something like this:
Mother Children
MotherID PK----\ ChildID PK
Mother Name \----> MotherID FK to Mother
Child Name