On Addition Theorems Related to Elliptic Integrals

We present formulas for the components of the Buchstaber formal group law and its exponent over $\mathbb Q[p_1,p_2,p_3,p_4]$. This leads to an addition theorem for the general elliptic integral $\int _0^x dt/R(t)$ with $R(t)=\sqrt {1+p_1t+p_2t^2+p_3t^3+p_4t^4}$. The study is motivated by Euler's addition theorem for elliptic integrals of the first kind.