A SIMPLE PROGRESSIVE TAX. 205 
the successive parts, is shown by the broken lines in the 
same diagram, the first line T = a,x (0 <x <x) being com- 
mon to both cases. 
It will be seen that in the first case T is a continuous 
function of x, and in the second case it is discontinuous at 
No, #1, X2, etc. 
§ 3. A natural extension of the scheme in the example 
at the beginning of $1 would be to let the equal increments 
come at intervals of £100, instead of at intervals of £500, 
the increments being proportionally reduced. In this way 
we would replace the sliding scale of that example by the 
following :— 
On the first £100, the rate shall be (53+ w) pence per £. 
On the second £100, the rate shall be (55 + a ) pence: per. 
On the third £100, the rate shall be (54+ = ) pence per £. 
On the twentieth £100, the rate shall be (53 + _ ) pence per £. 
And every pound over £2000 shall pay 93d. 
Then the reduction could be carried further, the incre- 
ments coming at each successive ae 
On the first £1, (54 + joo pence would be paid. 
On the second £1, (54 + 1000? pence would be paid. 
On the third IL 
n the thir £ Wh + 000) pence would be paid. 
|, 3999 
On the two thousandth Li, er os 7000 ) pence would be paid. 
And every Board over £2000 would pay 944d. 
But from the mathematical standpoint there is no reason why 
the reduction should stop at this stage. We might proceed to 
shillings, pence, and fractions of a penny, and in the limit we 
would reach a continuously increasing rate of tax from which the 
samme amount of tax would be obtained at the separate pounds. 
