= Strip density of a spherical system
{title2=$S(x)=2\pi\int_x^\infty r\rho(r)\,dr$}
The strip density is the mass per unit projected Cartesian coordinate $x$: $S(x)=\int_{-\infty}^{\infty}\Sigma(\sqrt{x^2+y^2})\,dy$. It is even in $x$. For $x\geq0$, changing variables to projected radius gives $S(x)=2\int_x^\infty\Sigma(R)R/\sqrt{R^2-x^2}\,dR$. Alternatively integrate the three-dimensional <mass density> over the plane of coordinates $y,z$ to obtain the displayed expression. Consequently $S'(x)=-2\pi x\rho(x)$ for $x>0$.
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