A course of pure mathematics, by G.H. Hardy.

FUNCTIONS OF REAL VARIABLES

27

upon unit area of the piston increases. Boyle's experimental law asserts that
the product of p and v is very nearly constant, a correspondence which, if
exact, would be represented by an equation of the type
pv =a.......................................(i),
where a is a number which can be determined approximately by experiment.
Boyle's law, however, only gives a reasonable approximation to the facts
provided the gas is not compressed too much. When v is decreased and p
increased beyond a certain point the relation between them is no longer
expressed with tolerable exactness by the equation (i). It is known that a
much better approximation to the true relation can then be found by means
of what is known as 'van der Waals' law,' expressed by the equation
( +) (V - y............................ (ii),
where a, 2, y are numbers which can also be determined approximately by
experiment.
Of course the two equations, even taken together, do not give anything
like a complete account of the relation between p and v. This relation is no
doubt in reality much more complicated, and its form changes, as v varies,
from a form nearly equivalent to (i) to a form nearly equivalent to (ii). But,
from a mathematical point of view, there is nothing to prevent us from contemplating an ideal state of things in which, for all values of v above a
certain limit, V say, (i) would be exactly true, and (ii) exactly true for all
values of v less than V. And then we might regard the two equations as
together defining p as a function of v. It is an example of a function which
for some values of v is defined by one formula and for other values of v is defined
by another.
This function possesses the characteristic (2): to any value of v only one
value of p corresponds: but it does not possess (1). For p is at any rate not
defined as a function of v for negative values of v; a negative volume means
nothing, and so negative values of v do not present themselves for consideration at all.
5. Suppose that a perfectly elastic ball is dropped (without rotation)
from a height ~gr2 on to a fixed horizontal plane, and rebounds continually.
The ordinary formulae of elementary dynamics, with which the reader is
probably familiar, show that
h= I gt2
if 0 < t,
h= g (2r-t)2
if r e t 3T, and generally
h=g (2n - t)2
if (2n-1)rT t (2n + 1)r, h being the depth of the ball, at time t, below its
original position. Obviously h is a function of t which is only defined for
positive values of t.
The reader should construct other examples of functions which occur in
physical problems.