which we will construct all of the strong-segregation phases. Our calculations are performed in terms of the ratio $$a(\lambda)\equiv {{\cal A\/}(z)\over {\cal A\/}(R)}\qquad(\lambda
=z/R) \label{eq:area}$$ of the cross-sectional area ${\cal A\/}$ at a height $z$ relative to that of the outer surface, in an infinitesimal wedge of height $R$. This function may be easily calculated for wedges of particular shapes by elementary geometry, and is collected in Table \[table1\] for various geometries. Since $a(\lambda)$ is the projected surface area along the normal vector extending from the wedge point to the flat wedge top, it will be a quadratic function of $\lambda$. The boundary condition $a(0)=0$ implies that $a(\lambda)$ is a sum of $\lambda$ and $\lambda^2$ terms, and the boundary condition $a(1)=1$ fixes the sum of the corresponding coefficients to be $1$, leaving a single parameter. Hence we may generally write $$a(\lambda) = p \lambda + (1-p) \lambda^2.
\label{eq:p}$$

The location $z_d$ of the “dividing surface” separating the two species is determined by equating the relative volume below $z$, denoted $v(z/R)$, to the volume fraction $\phi$: $$\begin{aligned}
v(z/R)&\equiv&\int_0^{z/R}\!dy\,a(y)
= {V(z)\over R {\cal A\/}(R)}\label{eq:vdef} \\
v(\beta) &=&\phi \, v(1) \quad \qquad (\beta=z_d/R) \quad , \label{eq:zdloc} \end{aligned}$$ where $V(z)$ is the (partial) volume 