1. If θ be the eccentric angle of a point P on an ellipse whose semi-axes are a and b and if x, y be the rectangular co-ordinates of a point Q, then PQ2 = (x − acosθ)2 + (y − bsinθ)2. When a = b and x2 + y2 = f2, PQ2 = f2 2afcosθ + a2, if the line from which θ is measured passes through Q. The integral

is a well-known one. I propose in the present paper to consider the more general form which the integral takes when P lies on an ellipse.