SEMANTICS OF THE DIFFERENTIAL CALCULUS 601 

In physics we deal with many processes which are structurally periodic, which means that a definite physical condition constantly recurs after equal intervals of 'time'. The number of seconds or fractions of seconds within which the process runs its course is called the period. We know already that the simplest periodic functions are sine and cosine functions of the type
tegral value. Furthermore we have already seen that the first derivatives, therefore all derivatives of such functions are likewise simple sine and cosine functions. In particular, the second derivatives of the sine and cosine functions are likewise sine and cosine functions taken with the opposite algebraic sign. If we express the variability of a process as a function of 'time', that is, by an equation of the form 5 = F(t), then in a periodic process,,
where T is the period and » any integer. If the process repeats itself, as in a periodic process, we must have 







but, as we have already seen, the sine and cosine functions satisfy these conditions.
A process which can be described by an equation of the type
is called a harmonic vibration or, a 'pure sine vibration', or simply a 'vibration' or 'oscillation'. The constant A, which represents the maximum value of the displacement on either side, is called the amplitude. The period T is called the 'time of vibration', its reciprocal value which gives the number of vibrations in a unit of 'time' is called the vibration number or frequency.
As the second derivatives of sine and cosine functions are equal to the original functions taken with the opposite signs, we can describe harmonic vibrations by differential equations of the first degree (linear) and of the second
order of the special type representing
the amplitude, T the period, e the phase of the vibration. The factor of proportionality a is taken as the square of any arbitrary real quantity to indicate that the righthand side must always have the opposite sign to that of S.
The propagation of a vibration is called an advancing plane wave which has both velocity and direction.
Fourier has shown that any given form of wave may be represented by the superposition of a series of sinewaves, which gives sinewaves great theoretical and practical importance.
In writing this chapter I had two main aims. One was to briefly indicate the essential semantic factors involved in the differential methods. The other, to make the general reader and even specialists who are not mathematicians acquainted with some terms and rudiments of method which will be necessary for further discussion. 
