342 
MR. C. GODFREY ON THE APPLICATION OF FOURIER’S 
waves is inconsiderable. The prepotent part of the energy resides in those quicker 
waves for which the energy curve is of the normal form (Compare the 
magnetic pulses of Professor Thomson, treated in the next chapter). 
§ 26. To recapitulate, the pulses of the sequence will not be separately dis¬ 
tinguishable ; their effect depends upon the integral of energy over an interval of 
time comparable with T ; the phase of the Fourier element will have no further 
effect; and all the observable properties of the sequence will be bound up with the 
energy function < p°(u ). 
§ 27. Hitherto it has been assumed that the interval T comprises a large number n 
of complete pulses, these being for the moment supposed not to be of infinite 
breadth. In general the boundaries of the interval T will find themselves in a pulse ; 
there will be a number of incomplete pulses near each end. But these are few com¬ 
pared with the whole number n ; they will not perceptibly affect the aggregate. 
The spectrum is independent of n as regards composition, if n is large. The total 
intensity, however, varies as n ; thus a variation in the crowding of the pulses 
causes a corresponding variation of the brightness of the spectrum—a result which 
might have been expected. 
§ 28. Let us consider how these results are affected when the individual pulses are 
of infinite breadth. Suppose that we examine the type which Lord Rayleigh 
suggested, 
f(t) = e-“. 
The displacement becomes comparatively small when the distance from the centre 
of the pulse is great compared with 1/c. In fact, we are tempted to regard these 
pulses as practically equivalent to pulses of finite breadth 1/c or thereabouts. 
Suppose that the least observable interval I 1 comprises a large number of central 
points of pulses. Suppose also that T is great compared with 1/c. The interval 
will contain a contribution from each of the infinite succession of pulses. But, since 
°o 
\e~ x \lx is small when b is great, only those pulses which contribute finite displace- 
b 
ments will affect the aggregate content of T. Now the centres of these will lie 
either in the interval, or at a distance from its extremities of order 1/c. As 1/c is 
small compared with T we shall practically be concerned only with the large number 
of pulses which lie almost entirely within T. We are, therefore, justified in 
regarding 1/c as the effective order of breadth of these strictly infinite pulses. 
§ 29. If the pulses (supposed finite) are crowded, so that they overlap largely, 
we shall not find the characteristic spectrum unless the time-interval T which we 
are investigating is large compared with the breadth of a pulse. If T cuts into many 
pulses, but is not large compared with the breadth of each, we shall lose sight of 
individual pulses, it is true; but the energy function will be largely affected by the 
incomplete pulses. In other words, if we can shorten the exposure till it is com¬ 
parable with the duration of a pulse, the spectrum observed will begin to show 
