Let $V$ be a variety of type $\tau$. A hypersubstitution 
which preserves all identities of $V$ is called a $V$-proper hypersubstitution.
The set $P(V)$ of all $V$-proper hypersubstitutions forms a monoid, 
which is a submonoid of the monoid of all hypersubstitutions of type $\tau$. 
The hypersubstitutions in $P(V)$ can be partitioned according 
to an equivalence relation $\sim_V$ first introduced by P \l onka. 
The authors introduce the name "degree of proper hypersubstitutions with respect 
to $V$" for the number $d_p(V)$ of distinct equivalence classes under this relation, 
and study the properties of this parameter.