Analytical solutions for three body motion are sparse in the literature but still sought for many reasons. In this paper, an analytic solution for motion of the third body in a circular orbit in the plane of motion of the two primaries is introduced, for small mass parameter and when the motion is in the vicinity of one of the two primaries. In this case, the Jacobi function allows implementation of Legendre polynomials, and the Jacobi integral equation is reduced to the Legendre normal form of an elliptic integral. A closed form expression for the period of motion is formulated and expressions for coordinates as functions of time are also introduced. The speed of the third body is found to be nonuniform along the path. The obtained sotution gives insights into the physics of the three body problem. This solution can be used as a generating orbit for purposes of numerical or analytical continuation of periodic orbits in three body systems in which mass parameter is close but not equal to zero.