It has already been indicated how the equations on which the arch theory is based may be simplified when the moment of inertia is constant.

The above problem was worked out on the basis that the moment of inertia varied in the ratio of ds/dx. In either case the solution is considerably simplified. Arches are frequently designed where the moment of inertia varies according to some other law. The very frequent practice is to increase the thickness of the arch toward the abutment much more rapidly than the ds/dx rule would call for, and thus increase the moment of inertia of the arch much more rapidly. In such a case, Equations 49 must be used; and the summations must be made up by computing for each unit-section the value of the moment of inertia for that point, and by measuring ds along the length of the arch rib. This means also that the sections of dead and live load, instead of having a constant width (as in the above problem), have a variable width, and the loads must be separately computed. While there is nothing especially difficult about such a solution, it involves considerably more work.