The exact solution recently obtained for the small-angle plural and multiple scattering of fast charged particles is used to derive several theorems of interest concerning scattering-produced curvatures, and calculations are reported of the probability distribution of such curvatures. Formulas are provided for obtaining the probability of occurrence of curvatures greater than any given amount, for any scattering material and energy, as long as the scattering is not too large. The types of curvature measurement dealt with are: (a) three-point observation, (b) mean curvature using the tangents to the track at each end, and (c) difference between the two curvatures obtained in (b). Asymptotic formulas allow the calculations to be extended for unusually rare events.

It is shown, first, that knowledge of the probability of a lateral displacement $x$, after a track length $z$, is sufficient to find the distribution of scattering-produced curvatures $c$ obtained by observing the ends of the track and any one interior point. The appropriate relationship is $x=\frac{z^{2}c}{2}$.

It is further shown that by introducing a lateral displacement angle $\phi =\frac{x}{z}$ the probability of observing $\phi$ in length $z$ may be obtained by a "duplication formula" from the distribution in ${\phi }_1$ at any $z_1<z$ and the distribution in $\phi -{\phi }_1$ in a track of length $z-z_1$.

From the basic correlated distribution formula, we have obtained distributions for the curvatures of the two circles tangent to either end of the track and passing through the other end. The mean $\overline{c}$ of these two curvatures (a special "four-point" curvature) follows the same distribution law as the directional distribution of the previous paper, with $\eta =z\overline{c}$. The difference of these curvatures, $D$, follows the same law as the three-point curvature $c$, above.

Finally, we have shown that the distribution in $x$ and $\eta$, or in $c$ and $\overline{c}$, are nearly the same if we write $\eta =\sqrt{3}\left(\frac{x}{z}\right)$ or $\overline{c}=\frac{\sqrt{3}c}{2}$, although the exact solution yields different expressions for the appropriate Fourier transforms.

Calculations are here reported and summarized in graphs of the differential and integral distributions in $\phi =\frac{x}{z}$, in terms of dimensionless units $\frac{z}{\lambda }$ and $\frac{\phi }{{\eta }_0}$ as in the previous work. Integral distributions in angular displacement, $\eta$, derived from the previous calculations, are included for completeness.

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