The author derives and proves a formula for the Donaldson invariant of a 4-manifold evaluated on the product of a high power of a spherical class with a perpendicular homology class. The formula is valid provided the sphere has negative self-intersection, say \(-p\). The invariant is expressed in terms of invariants of products of the perpendicular class with low powers of the spherical class \((\leq p)\) and with powers of the point class. Results in Donaldson theory tend to be overlooked after the introduction of Seiberg-Witten theory. This is because Seiberg-Witten theory is technically less complicated, yet is conjectured to provide the same information. Results similar to the results in this paper may be used to compute the complete Donaldson invariants for large classes of 4-manifolds, thereby proving the equivalence of Seiberg-Witten theory and Donaldson theory for these classes. In addition, the simple type conjecture has not been proved, and it is possible that techniques similar to the techniques used in this paper could resolve the simple type conjecture. The main theorem of this paper is proved by a neck-stretching argument. The paper gives a very detailed account of the geometric arguments involved and clearly states and gives references to the analytical gluing results used.