Here's an alternative derivation of Eq. 3.10 (the surface charge density induced on a grounded conducted plane by a point charge q a distance d above the plane). This approach (which generalizes to many other problems) does not rely on the method of images. The total field is due in part to q, and in part to the induced surface charge. Write down the z components of these fields—in terms of q and the as-yet-unknown σ(x,y)—just below the surface. The sum must be zero, of course, because this is inside a conductor. Use that to determine σ.