A formula relating the level shifts and widths to the meson-nucleus scattering lengths is examined for a momentum-dependent potential of general form, using a simple argument similar to that of Bethe and deHoffmann and Coulomb effective-range theory.

Expressions are given for the electromagnetic final-state interaction contribution to $D$, the triple-product correlation coefficient, in the nuclear $\beta$-decay transitions ½ → ½, $\frac{3}{2}\to \frac{3}{2}$, ${\frac{3}{2}}^{\pm }\to {\frac{1}{2}}^{\pm }$, ${\frac{5}{2}}^{\pm }\to {\frac{3}{2}}^{\pm }$, and $1^{+}$→$0^{+}$ to all orders of $\frac{Z\alpha }{v_e}$ in the Coulomb interaction.

The reaction ${\pi }^{-}p\to \eta n$ has been observed through the decay mode $\eta \to 2\gamma$ at $T_{{\pi }^{-}}=592, 655, 704, 875, 975, 1117, \mathrm{and} 1300$ MeV. The detection apparatus was a cubic array of six steel-plate optical spark chambers that completely surrounded a liquid-hydrogen target. We identified events attributed to the decay of an $\eta$ by the large c.m. opening angle between the two showers generated in the steel plates by the decay photons. We have calculated the total cross section for $\eta$ production, which is proportional to the number of events under the large-angle peak in the opening-angle distribution. The total cross section rises steeply from threshold to a maximum of about 2.4 mb at 650 MeV, and then falls gradually to about 0.66 mb at 1300 MeV. The differential cross section was obtained by taking the coefficients of a Legendre-polynomial fit to the angular distribution of bisectors of selected two-shower events, and converting them to the coefficients of the $\eta$ c.m. angular distribution. The differential cross section is found to be isotropic at 592 MeV, to require terms through $P_2\left({cos\theta }_{\eta }\right)$ between 655 and 975 MeV, and to have a forward peak fitted by terms through $P_3\left({cos\theta }_{\eta }\right)$ at 1117 MeV and through $P_4\left({cos\theta }_{\eta }\right)$ at 1300 MeV. It is suggested that production at threshold is predominantly through an $S$ state, with some $P_1$ and $D_3$ waves entering by 655 MeV. We suggest that all the absorption in the $S_{11}$ $\pi N$ state can be explained by the $\eta$ production.

A general phenomenological method for studying a two-pion mass spectrum is developed; it is shown that, without assumptions about the production mechanism of the two-pion system, no significant upper limit on the $\omega \to {\pi }^{+}{\pi }^{-}$ branching ratio can be set with present-day experiments. In general, a lower limit may be set if a significant effect is seen. This method is applied to data containing more than $8000\omega \to {\pi }^{+}{\pi }^{-}{\pi }^0$ events. The lower-momentum half of the sample, which shows a significant $\omega \to {\pi }^{+}{\pi }^{-}$ signal, was published previously but is here reanalyzed to set a lower limit on the $\omega \to {\pi }^{+}{\pi }^{-}$ branching ratio. The new data at higher momenta show no significant $\omega \to {\pi }^{+}{\pi }^{-}$ signal. The results from the various momenta are shown to be consistent.