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Prove that for the curve
\( s=m\left(\sec ^{3} \psi-1\right), \rho=3 m \tan \psi \sec ^{3} \psi \),
and hence that \( 3 m \frac{d y}{d x} \frac{d^{2} y}{d x^{2}}=1 \).
Also, that this differential equation is satisfied by the semicubical parabola
\[
27 m y^{2}=8 x^{3}
\]
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