  The present paper aims at describing in details the asymptotic composition of
a class of d-colour P\'olya urn: namely balanced, tenable and irreducible urns.
We decompose the composition vector of such urns according to the Jordan
decomposition of their replacement matrix. The projections of the composition
vector onto the so-called small Jordan spaces are known to be asymptotically
gaussian, but the asymptotic behaviour of the projections onto the large Jordan
spaces are not known in full details up to now and are discribed by a limiting
random variables called W, depending on the parameters of the urn.
