Formulae are given for the coefficients of the highest powers of \(\lambda\) in the chromatic polynomial \(P(H,\lambda)\) of a linear uniform \(h\)-hypergraph \(H\), thus generalizing the corresponding result of \textit{G. H. J. Meredith} for graphs [J. Comb. Theory, Ser. B 13, 14-17 (1972; Zbl 0218.05056)]. Some differences appear whenever (\(g= 3\), \(h= 3\)); \((g=4\), \(h=3\)) or (\(g=3\), \(h= 4\)), where \(g\) is the girth of the hypergraph \(H\); in this case we must count the number of subhypergraphs of \(H\) from a list of thirteen hypergraphs given in the paper, seven of these being subhypergraphs of the Fano configuration. It is proved that if simple \(h\)-hypergraphs \(H\) and \(G\) are chromatically equivalent and \(H\) is linear then \(G\) is linear, too. Also, if \((g,h)\neq (3,3)\) then two chromatically equivalent \(h\)-hypergraphs must have the same order, size, girth and number of \(g\)-cycles. It is proved that the elementary \(h\)-uniform cycle \(C^h_m\) with \(m\) edges is chromatically unique for all \(m,h\geq 3\); the same property possesses a kind of bicycles defined in the paper.