Preliminary Definitions

A variable is useful if it occurs in the derivation of some string. This requires that

the variable occurs in some sentential form (you can get to the variable if you start from S), and
a string of terminals can be derived from the sentential form (the variable isn't a "dead end").

A variable is recursive if it can generate a string containing itself. For example, variable A is recursive if S

uAy for some values of u and y.

A recursive variable A can be either

directly recursive, that is, there is a production Ax₁Ax₂ for some strings x₁, x₂ (T V)*, or
indirectly recursive, that is, there is are variables X_i and productions
A...X₁...
X₁...X₂...
X₂...X₃...
...
X_N...A...