340 
MR. C. GODFREY ON THE APPLICATION OF FOURIER’S 
an obvious condition is that the pulses shall not be so far apart as to he separately dis¬ 
tinguishable. Photography can fix 10“ 7 second ; hence there must be many pulses in 
10 -7 second. This is a condition certainly fulfilled by Rontgen rays. The coarser 
the means of observation which we use, the more thinly may the pulses be scattered. 
The results which we are about to investigate may be true, for a certain radiation, in 
the present state of experimental science ; but will cease to be true for that particular 
kind of radiation when our instrumental means shall have been so improved as to 
enable us to distinguish structure in that radiation. 
It shall be shown that this is the only condition necessary in order that a random 
sequence of similar pulses may be equivalent to radiation of a spectral composition 
given by the analysis of a single pulse. 
§ 24. We have already proved that we are. concerned simply with an integral effect 
over a time T of the order of the shortest observable interval. If we are content to 
view the radiation with the eye, or to use a slow photographic plate, T may be taken 
as great as we please. If, on the other hand, we are investigating the radiation 
with the shortest possible exposure, T may be reduced as far as our experimental 
skill will allow. 
Let us examine the Fourier composition of a numerous sequence of random similar 
Suppose the pulse to be 
oo 
f(i ■) = ( cos (ut -f- \jf)clu 
b) 
the angle xfj being a definite function of u. We are to examine the Fourier integral, 
equivalent to 
f{* ~ P) +f(t ~ To) +f(t - r 3 ) + . . . + f(t - r«) 
where t,, t 2 , t 3 , . . . r n define a large number of points of time distributed at 
random in an interval T. The breadth of the pulse is to be small compared with T. 
The resultant integral is 
co 
[ (f)(u){cosut — ut j fi- i f) -f cos ut — ht. : + )//... -j- cos ut — ut„ + xfs}du. 
Jo 
Consider the quantity 
< j>(u){cOSUt + xfj — UT X + COS ut -F xfj — UT g COS Ut + l ff — UT„} . . A. 
First, suppose that the time of vibration, 2tt/u, is small compared with T or r„ — r t . 
Draw from an origin lines of length making with the prime vector angles 
x/j — ut v xfj — ut 2> &c. Then the bounding lines of the angles xp—UT ]} xfj — ut. 2 , . . , 
