  This work explores simple relations that follow from the momentum-integral
equation absent a pressure gradient. The resulting expressions enable us to
relate the boundary-layer characteristics of a velocity profile, $u(y)$, to an
assumed flow function and its wall derivative relative to the wall-normal
coordinate, $y$. Consequently, disturbance, displacement, and momentum
thicknesses, as well as skin friction and drag coefficients, which are
typically evaluated and tabulated in classical monographs, can be readily
determined for a given profile, $F(\xi)=u/U$. Here $\xi=y/\delta$ denotes the
boundary-layer coordinate. These expressions are then employed to provide a
rational explanation for the 1921 Pohlhausen polynomial paradox, namely, the
reason why a quartic representation of the velocity leads to less accurate
predictions of the disturbance, displacement, and momentum thicknesses than
using cubic or quadratic polynomials. Not only do we identify the factors
underlying this behaviour, we proceed to outline a procedure to overcome its
manifestation at any order. This enables us to derive optimal piecewise
approximations that do not suffer from the particular limitations affecting
Pohlhausen's $F(\xi)=2\xi-2\xi^3 +\xi^4$. For example, our alternative profile,
$F(\xi)=(5\xi-3\xi^3+\xi^4)/3$, leads to an order-of-magnitude improvement in
precision when incorporated into the K\'arm\'an-Pohlhausen approach in both
viscous and thermal analyses. Then noting the significance of the Blasius
constant, $\bar{s} \approx 1.630398$, this approach is extended to construct a
set of uniformly valid solutions, including
$F(\xi)=1-\exp[-\bar{s}\xi(1+\bar{s}\xi/2+\xi^2)]$, which continues to hold
beyond the boundary-layer edge as $y\rightarrow\infty$. Given its substantially
reduced error, the latter is shown, through comparisons to other models, to be
practically equivalent to the Blasius solution.
