It has been pointed out that Mayer's method of solving the differential equation

is quite general and only requires one integration, whereas the other general methods require two, or even thee, integrations. The reason for this is rather obscure in most text-books ; and the proof of the validity of the method rests on an existence theorem which leads to a solution of an apparently different form. I shall prove here that the two forms are identical.

Consider the functional equation

Now let and These are elements of a group, G1 say, which is isomorphic to C2, the cyclic group of order 2, as can be seen from the group table:

In every textbook which deals with second order linear finite-difference equations, methods will be found for solving such equations when the coefficients are constants. In the author’s experience it is unknown for textbooks to treat second order linear difference equations with variable coefficients. Yet such equations certainly occur in practical applications. It is the present intention to remedy this defect in the literature. It will be seen that, by the methods discussed here, we can always reduce the solution of a linear second order difference equation to that of a first order equation.

The purpose of this note is to show how two first integrals of the equation of motion of a particle moving under the action of an inverse square central force can be obtained very simply using vector techniques. In fact it is shown that, amongst all central forces, only the physically important inverse square force can be treated using the technique given here, thus demonstrating one facet of the mathematical simplicity of that force.