DOUBLE INTEGRALS TO OPTICAL PROBLEMS. 
339 
relations disappear, and we are left with the energy curve to completely specify the 
properties of the sequence of pulses. These statements will be justified below. 
§21. The subject has been opened by Lord Rayleigh* in his paper on “The 
Complete Radiation at a given Temperature.” He proposes to regard this as an 
irregular sequence of pulses of the type e~" 
Now 
e 
-cV 
CO 
= M 
C7T~ Jn 
cos v.rdu 
( 8 ) 
and the whole energy of the pulse 
~**dx == \ [ c lfi 2 '°du 
c- J 0 
(22) 
The intensity corresponding to the limits u and u + du is therefore c~ 2 e~ u%, ~ c *du. 
“ If an infinite number of impulses, similar hut not necessarily equal to (8), and of 
arbitrary sign, be distributed at random over the whole range from — oo to + °°, the 
intensity of the resultant for an absolutely definite value of u would be indeter¬ 
minate. Only the probabilities of various resultants could be assigned; and if the 
value of u were changed, by however little, the resultant would again be indeter¬ 
minate. Within the smallest assignable range of u there is room for an infinite 
number of independent combinations. We are thus concerned only with an average, 
and the intensity of each component may be taken to he proportional to the total 
number of impulses (if equal) without regard to their phase-relations. In the aggre¬ 
gate vibrations, the law according to which the energy is distributed is still, for all 
practical purposes, that expressed by (22).’’ 
§ 22. This important paragraph suggests the whole theory. But when we come to 
take a closer view of it, it will be found that there are certain questions which still 
remain to be solved. 
Suppose, for instance, that the elementary pulse is confined to a certain small 
range of time, such as are the pulses in Professor Thomson’s theory, how 
many of those pulses must be present in order to give the properties which Lord 
Rayleigh associates with an infinite succession ? Again, we might suppose that 
different consequences would follow from different degrees of crowding among the 
pulses. They may be so close, on the average, that a great number of them are 
everywhere found overlapping ; or, again, they may be so thinly scattered as to be, 
on the average, far apart in comparison with the space occupied by each. Experiment 
does not allow us to say which of these suppositions is correct; we may enquire what 
is the test which determines the applicability of Lord Rayleigh’s theorem to the 
aggregate. 
§ 23, In order that the sequence of pulses may, for us, he equivalent to a spectrum, 
* ‘ Phil. Mag.,’ 27, 1889. 
2x2 
