  We show that choreographic three bodies {x(t), x(t+T/3), x(t-T/3)} of period
T on the lemniscate, x(t) = (x-hat+y-hat cn(t))sn(t)/(1+cn^2(t)) parameterized
by the Jacobi's elliptic functions sn and cn with modulus k^2 = (2+sqrt{3})/4,
conserve the center of mass and the angular momentum, where x-hat and y-hat are
the orthogonal unit vectors defining the plane of the motion. They also
conserve the moment of inertia, the kinetic energy, the sum of square of the
curvature, the product of distance and the sum of square of distance between
bodies. We find that they satisfy the equation of motion under the potential
energy sum_{i<j}(1/2 ln r_{ij} -sqrt{3}/24 r_{ij}^2) or sum_{i<j}1/2 ln r_{ij}
-sum_{i}sqrt{3}/8 r_{i}^2, where r_{ij} the distance between the body i and j,
and r_{i} the distance from the origin. The first term of the potential
energies is the Newton's gravity in two dimensions but the second term is the
mutual repulsive force or a repulsive force from the origin, respectively.
Then, geometric construction methods for the positions of the choreographic
three bodies are given.
