PDE — characteristic curves and envelopes

Consider the statements: (I) The characteristic curve of a one parameter family of surfaces $f(x,y,z,a)=0$ is a locus of the common points of intersection of $f=0$ and $\partial f/\partial a=0$. (II) The envelope of the one parameter family $f(x,y,z,a)=0$ is obtained by eliminating $a$ from $f=0$ and $\partial f/\partial a=0$. Then: