  The Blaschke-Petkantschin formula is a variant of the polar decomposition of
the $k$-fold Lebesgue measure on $\mathbb {R}^n$ in terms of the corresponding
measures on $k$-dimensional linear subspaces of $\mathbb {R}^n$. We suggest a
new elementary proof of this formula and discuss its connection with the
celebrated Drury's identity that plays a key role in the study of mapping
properties of the Radon-John $k$-plane transforms. We give a new derivation of
this identity and provide it with precise information about constant factors
and the class of admissible functions.
