In 1901, Hilbert posed 23 mathematical problems of which the second part of the 16 th o ne is to find the maximal number and relative positions of limit cycles of planar polynomial systems of degree n = (x,y), = (x,y). There are many works on finding the maximal number of limit cycles and raising the lower bounds of Hilbert number H(n) for general planar polynomial systems or for individual degree of systems. A detailed introduction and related literatures can be found in Li, Schlomiuk, Ilyashenko, and Han. Many studies had been done for planar systems close to Hamiltonian systems, especially for quadratic and cubic systems. The main results are on the number of limit cycles which appear near a center, a periodic annular or a homoclinic loop by perturbations. The first order Melnikov function which is called also and Abelian integral plays and important role in getting these results. It can be used to study Hopf, homoclinic and heteroclinic bifurcations, and the so-called weak Hilbert’s 16 th problem as well. It has been shown that equivariant cubic systems can have 12 small amplitude limit cycles, and that equivariant cubic systems can have 4 limit cycles. Here we study the analytic properties of the first Melnikov function for general near-Hamiltonian systems exhibiting a cuspidal loop for order m and obtain its asymptotic expansion at the Hamiltonian value corresponding to the loop. More precisely, we suppose that the unperturbed system = , =- ,