206 H. S. CARSLAW. 
In this example, the continuous rate would be given by the 
y 
equation Se = 5) + “s a 
1 
and we = (0. 
and we would have 7’ = (54 + 1000 x) x 
If the figures had been (a + 4 6) for the first £1, (a + % 6) for 
the second £1, and so on, we would have been led to the continu- 
aT 
ous rate =ad¢+ ba, 
a x 
and 7’ = (a + 3 b«x)za. 
§ 4. So far as I know, the Commonwealth of Australia 
was the first to introduce a sliding scale in which the 
amounts paid on each successive pound—at least up to a 
certain point—form an arithmetical progression, while 
every pound over that point pays the same, namely, at the 
dT 
(ane 
rate ( Ad 
the increments occur. 
According to the Federal Income Tax Act, 1917, the 
amount of the tax on an income derived wholly from per- 
sonal exertion may be calculated as follows:* 
(i.) When the whole taxable income does not exceed £7600, the 
3 
800 
\ reached at the end of the last interval in which 
amount of the tax on a taxable income of £x shall be (3 + —_ax)ax 
pence. 
(11.) When the whole taxable income exceeds £7600, the amount 
of the tax on the first £7600 shall be £997 10s., and every pound 
over £7600 shall pay 5s. 
It is easy to show that this is equivalent to the following 
scheme :— 
On the first £1, (3 + 7 pence shall be paid. 
On the second £1, (3 + ae ) pence shall be paid. 
" In the First Schedule of the Act another form of words is used, but 
the fundamental clauses lead to the above result. This system was intro- 
duced when the Federal Parliament first imposed an Income Tax in 1915. 
There is now a considerable super-tax. 
