In an experiment at the Argonne Zero-Gradient Synchrotron we have measured values of the polarization parameter $P\left(t\right)\mathrm{}$ in the elastic scattering of negative pions, positive pions, positive kaons, and protons on protons at several incident laboratory momenta from 2.50 to 5.15 $\frac{\mathrm{GeV}}{c}$, and for values of the momentum transfer variable $-t$ from 0.2 to 2.0 ${\left(\frac{\mathrm{GeV}}{c}\right)}^2$. The final results from $p-p$ elastic scattering presented here extend our knowledge of the polarization to much larger values of $-t$ than the results of previous measurements. Outstanding features revealed by these polarization data include (1) the development of a dip at about $-t=0.7\mathrm{} {\left(\frac{\mathrm{GeV}}{c}\right)}^2$, with (2) a substantial secondary peak at larger values of $-t$ and (3) the gradual diminution of the maximum polarization with increasing energy. It is possible to fit the $t$ dependence of the experimental results with a simple model. The energy dependence of the polarized cross sections is also discussed.

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