= Solution
For an axisymmetric <viscous gravity current on a cone>, the cylindrical radius at slant distance $x$ is $x\cos\alpha$. A surface annulus therefore has area $2\pi x\cos\alpha\,dx$. The local downslope <volume flux> is the same as in part (a), but <conservation of mass> must account for this growing circumference. Thus
$$
\boxed{h_t+\frac Kx\partial_x\left[xh^3(\sin\alpha-\cos\alpha\,h_x)\right]=0,
\qquad V=2\pi\cos\alpha\int_0^{x_N(t)}xh\,dx.}
$$
This is a leading thin-film description away from the apex, with $0<\alpha<\pi/2$ and thickness small relative to the local curvature and streamwise length scales.
The original PDF uses the long-time criterion $x_N\gg(V\csc\alpha)^{1/3}$; the local TeX incorrectly reads $V\cos\alpha$. To see why the PDF criterion is the relevant one, scale $h\sim V/(x_N^2\cos\alpha)$. In the bulk, the hydrostatic-gradient term relative to direct downslope gravity is
$$
\cot\alpha\,|h_x|\sim\frac{V}{x_N^3\sin\alpha}\ll1.
$$
Hence the leading outer evolution, with $\beta=K\sin\alpha$, is
$$
h_t+\frac\beta x\partial_x(xh^3)=0.
$$
For a fixed-volume <similarity solution>, let $x_N\propto t^b$ and $h\propto t^a$. <Conservation of mass> gives $a+2b=0$, while the evolution equation gives $a-1=3a-b$. Thus $b=1/5$, $a=-2/5$. Writing $h=t^{-2/5}H(\xi)$, $\xi=x/t^{1/5}$, yields
$$
-\frac25H-\frac15\xi H'+\frac\beta\xi(\xi H^3)'=0.
$$
The positive branch $H=c\sqrt\xi$ satisfies this equation when $c^2=1/(5\beta)$. It gives
$$
\boxed{h(x,t)=\sqrt{\frac{x}{5\beta t}},\qquad 0<x<x_N(t),\qquad
\beta=\frac{\rho g\sin\alpha}{3\mu}.}
$$
The ideal outer current has zero thickness beyond its front. Its mass integral is
$$
V=\frac{2\pi\cos\alpha}{\sqrt{5\beta t}}\int_0^{x_N}x^{3/2}\,dx
=\frac{4\pi\cos\alpha}{5\sqrt{5\beta t}}x_N^{5/2}.
$$
Therefore
$$
\boxed{x_N(t)=\left(\frac{125\beta V^2t}{16\pi^2\cos^2\alpha}\right)^{1/5}
=\left(\frac{125\rho g\sin\alpha\,V^2t}{48\pi^2\mu\cos^2\alpha}\right)^{1/5}.}
$$
The reduced equation is a <conservation law> after multiplication by $x$. Its front obeys the <Rankine-Hugoniot condition>, $\dot x_N=\beta h_N^2=x_N/(5t)$, consistent with the mass-derived front position. The outer profile has a finite jump at the front; the omitted hydrostatic-gradient term resolves a narrow front region. There is also a small apex region where the outer gradient and conical curvature violate the local approximations. Neither region sets the leading conserved-volume scaling above.
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