336 
MR. C. GODFREY OX THE APPLICATION OF FOURIER’S 
Interference with given path difference 2T. 
This depends on 
+f(t + 2r)j 2 - f(l) -f-(t + 2t)] = i \+ 2r )dt. 
Now 
f(f) = ( R cos (at + \jj)du 
J o 
CO 
f(t ff- 2t) = ( Rcos(w/ -f- 2 ut + xjj)clu, 
Jo 
+ 00 CO 
f f(t)f(t + 2r)(A = 771 R 2 cos 2uTdtt. 
J -'o 
— co 
The phase i jj has disappeared. # 
Influence of Light on a Vibrator. 
The phenomena of refraction, dispersion and absorption, can be explained (subject 
to the above reservation) by considering the action of light waves on a vibrator. 
The equation of motion of the vibrator is 
co 
x + 2 kx + grx = f{t) — I R cos (ut + xjj)du 
Jo 
CO 
= real part off Re :itU+,j) du. 
J o 
The “ general ” solution of this equation will be of the form Ae~ H cos (rt -f- 6 ), 
where r 2 = p 1 — /,: 2 and A, I are arbitrary. The above solution is to hold for all 
time from — co to + co. We must therefore put A = 0, and the complete solution 
will be 
x 
. „ f Un¬ 
real part ol —-- 
J o T— u 
E eW+Vdiu 
+ 2 kiu * 
CO 
=1 
lldu 
(f - uf + 4 kht, 
I d(l 2 ~~ w 3 )cos (ut + i ft) + 2 ku sin (vt -f iff)}. 
The average energy of the motion excited will depend upon 
f R 2 d% 
Jo (p~ — + Vru~ 
Again, the work done by the light depends upon \f(t)xdt. Applying Schuster’s 
* Schuster, £ Phil. Mag.,’ vol. 37, p. 533. 
t We verify that this is a solution. This involves the process of differentiating inside the integral. 
Now, the condition that a Fourier integral shall admit of being so treated is, that the function repre¬ 
sented shall be free from discontinuities and shall vanish at ± qo . No mathematical discontinuities will 
occur in a physical problem : and, if necessary, the conditions at infinity may be satisfied by introducing 
into /(/) a factor (such as er aH% where a is small) which shall ensure dying away at both extremities, and 
which at the same time will not affect the Fourier resolution in any marked degree. 
