In this paper I propose to describe a scheme of progressive taxation which has been introduced in Australia in the “Income-Tax Act, 1915” (Commonwealth of Australia). As a novel illustration of the use of the Integral Calculus it may be of interest to readers of the Mathematical Gazette. Incidentally I desire to point out the need for care in using the term “rate of tax” when such a progressive tax is described. In the schedules of this Act the words are used in at least two different senses. And if the curves of the second degree, or any degree, are to be referred to in Acts of Parliament—a step the wisdom of which certainly may be questioned so long as the mathematical knowledge of the average man remains what it is—the nature of the curves might with advantage be more clearly stated than is done in this Act.

There are two ways in which the solution of a particular linear differential equation may “fail” although the solulion of a more general equation obtained by replacing certain constants by parameters is complete.

where D as usual stands for d/dx.

For the general equation

the perfectly general solution is

A, B being independent arbitrary constants, but if we attempt to apply this solution to the particular equation (l), we find in the first place that the coincidence of n with l and m renders the first term infinite, and in the second place that the coincidence of m with l leaves us with only one effective constant, A + B. The method by which in the commoner textbooks the passage from the general solution to that of a particular equation is made in such cases as this is unconvincing.