Since $$\text{cn}^2 u+\text{sn}^2 u=1,$$
$$x=r\ \text{cn}\ u,$$
$$y=r\ \text{sn}\ u,$$
where sn and cn are sinus amplitudinis and cosinus amplitudinis,
yields
$$x^2 + y^2 =r^2.$$
Using Jacobi's ellipse, we can now construct the following integral.
$$w=\int_0^u \frac{d\mu}{\sqrt{1-l^2\ \text{sn}^2 \mu}},$$
which satisfies the differential equation associated with the algebraic curve
$$y^2=x^6 + ax^2 + bx + c$$
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