You piqued my curiosity, so I sat to play a little with the equation. If R is the usual radial coordinate, I got:
dh/dR = c y^(1/3) / [(1 - c^2 y^(2/3))^(1/2)]
with y = 3h/(2c), using the definition of c in my previous comment. (I got this by switching from dh/dr to dh/dR with the relation between sin and tan, and replaciong r by r(h). Feel free to check my math, I might have made a mistake.) This was easily integrated by Mathematica, giving a result that is too long to write here, but has no particular problem at h = 0, other than dR/Dh being infinite there. That is expected, it just means dh/dR = 0 so the peak of the dome is a smooth maximum.
You piqued my curiosity, so I sat to play a little with the equation. If R is the usual radial coordinate, I got:
dh/dR = c y^(1/3) / [(1 - c^2 y^(2/3))^(1/2)]
with y = 3h/(2c), using the definition of c in my previous comment. (I got this by switching from dh/dr to dh/dR with the relation between sin and tan, and replaciong r by r(h). Feel free to check my math, I might have made a mistake.) This was easily integrated by Mathematica, giving a result that is too long to write here, but has no particular problem at h = 0, other than dR/Dh being infinite there. That is expected, it just means dh/dR = 0 so the peak of the dome is a smooth maximum.