DOUBLE INTEGRALS TO OPTICAL PROBLEMS. 
Of) 
he really considers the effect of only a single waxing and waning of the incident 
light. It is true that, on the average, the amplitude of the sum of a large number 
of vectors of random phase is small compared with the sum of the amplitudes. 
At the same time the energy is, on the average, equal to the sum of the component 
energies. In the present case the right deduction is, that the energy of the natural 
vibration will vary with the number of the component vibrations ; in other words, 
will vary as the time elapsed since the light began to act. It will become greater 
without limit. It is easily seen from Sellmeier’s analysis that there is no tendency 
for the natural vibrations excited by successive fluctuation to counteract one another. 
As regards the forced vibrations, on the other hand, the phases are, so to speak, 
arranged so that there shall be no accumulation of energy. 
§ 69. For a frictionless vibrator, then, common homogeneous light will give a 
continually-increasing motion in the natural mode; this feature being entirely due 
to the irregularities. 
But if there were a friction, however small, the motion would be prevented from 
mounting up indefinitely. The vibrations started by the component trains would 
not persist; in fact we may expect to find that the natural vibration settles down to 
a definite state, depending on the damping and the nature of the irregularities. 
§ 70. The whole of this matter becomes quite simple on the application of Fourier. 
Let us first try to solve 
CO 
• • f 
x + p~x = l Bcos(i«f -f- xp)dn. 
Jo 
W e are tempted to take for solution 
f RcosN tff fa. 
Jo P'~ ~ u ~ 
This expression, however, has no definite value. The integrand involves an 
infinity at u = p ; furthermore, the infinity is of such a nature that the integral 
Jo + ! 
, / ' 
depends upon e/e, 
The above integral, in fact, is not a solution of the equation. We are forced to 
include a frictional term in the equation. But this corresponds to the actual pro¬ 
perties of the vibrator; we have shown that, but for damping, the natural vibration 
would continually increase; a state of things unknown among the observed effects 
of light. 
§ 71. On page 336 it has been shown that light of composition JB "?du will excite 
vibration of composition 
f 
Urdu 
u 2 ) 3 + 4 K~U~ 
3 A 
VOL. CXCV.-A. 
