Ulysses data in the high-speed solar wind that cover a wide range of latitudes centered on the solar poles were studied to test whether a polytrope law can be used to close the ion energy equation. Three approaches were taken. We determined the correlation between proton temperature and density (1) in the free expansion of the high-speed solar wind between 1.5 and 4.8 AU, (2) in steepened microstreams at high latitudes, and (3) at the edges of the equatorial band of solar wind variability. Strong correlations were observed in all data subsets that are consistent with a single polytrope relation, Tp=aNp(&ggr;*-1), where our best estimate for &ggr;* is between 1.5 and 1.7. The best fitting relation is Tp=(2.0¿0.13)¿105 Np0.57. ¿ 1998 American Geophysical Union