We show that the sequence spaces $a^r_0$, $a^r_c$ and $a^r_{\infty}$ are equal to the sets of all sequences whose Ces\`aro means of order 1 converge to 0, converge and are bounded. As a consequence of this, we are able to considerably simplify the known results and their proofs in [1, 2], and to add the characterisations of some more classes of matrix transformations. Chandrasekhara Rao 249