On the basis of analyticity alone it is shown that in the high-energy limit the partial-wave amplitude $a_l\mathrm{}\left(s\right)\mathrm{}$ for $l>\mathrm{}\mathrm{const}.{\ln }^{2}s$ do not contribute substantially to the scattering amplitude for finite angles. For an angle depending upon $s$ this result can be extended to show that the contribution to the scattering amplitude $T[s, cos\theta \mathrm{}(s\left)\right]$ from $a_l\mathrm{}\left(s\right)\mathrm{}$ for $l>\mathrm{}\frac{\mathrm{const}.{\ln }^{2}s}{sin\theta \mathrm{}\left(s\right)\mathrm{}}$ is smaller than any power of $s$. Here $s$ and $\theta$ are the square of the energy and the scattering angle in the center-of-mass system. Classically, this means that a bombarding particle whose impact parameter is larger than the shrinking quantity $\frac{\mathrm{const}.{\ln }^{2}s}{s^{\frac{1}{2}}sin\theta \mathrm{}\left(s\right)\mathrm{}}$ cannot be appreciably scattered through an angle $\theta \mathrm{}\left(s\right)\mathrm{}$. It is necessary to use the unitarity condition only when this result is used to put an upper bound on $T[s, cos\theta \mathrm{}(s\left)\right]$. Here $T(s, cos\theta )$ is assumed to be analytic as a function of $cos\theta$ in an $s$-independent complex neighborhood of the real segment (-1, 1) except for its intersection with the cuts from $\infty$ to $x\left(s\right)\mathrm{}$ and from $-x\left(s\right)\mathrm{}$ to $-\infty$, $x\left(s\right)\mathrm{}$ being an arbitrary function with $x\left(s\right)>1\mathrm{}$. Finally the forward and backward peaks are discussed and it is proved that the right- (left-) hand cut contributes only to the forward (backward) peak and not to the backward (forward) peak.

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