200 
Proceedings of the Royal Society of Edinburgh. [Sess. 
importance are those for which a is nearly equal to /3. We can therefore 
neglect the second term entirely. 
We then find by the aid of a theorem given by Lord Rayleigh * that 
the energy of the waves lying between a and a -{-da is given by 
1 sin 2 a — fi)l 7 
7T (a — ft)' 1 
This expression obviously has a maximum value for a = /3, and falls away 
rapidly on both sides of this maximum to a minimum value, zero, given by 
= ±27r. It has a succession of minima given by (a — /3)£ = ±2^71" , 
where n is any integer. Only the central maximum is of importance ; it is 
twenty-one times as high as those on each side of it. The expression, in 
fact, is the same as that representing the diffraction pattern seen when an 
unlimited plane harmonic wave falls normally on a slit, and, as such, it is 
discussed in most books on optics. 
Let us suppose, now, that the original wave is received by a spectroscope 
of infinite resolving power and with an infinitely narrow slit. It does not 
converge to a mathematical line at the point in the spectrum specified by 
/3, as it would do if it were an infinitely long train, but, owing to its being 
limited, forms a maximum of finite breadth at this point, accompanied by 
fainter lines arranged symmetrically on each side. 
Let the initial wave be given by e~ ht cos /3t from t — 0 to t = co ; i.e. it 
starts with a finite value, and slowly damps down to zero. Then the 
Fourier integral becomes 
if da [ e -/t Hos /3£ cos a($- t)d£= ~ f da f e~ h % cos{(a - f$)£— at}d£, 
W 0 Jo AT?) 0 J 0 
since the other term can be neglected as in the previous case. On integrat- 
ing and substituting the limits this becomes 
1 r coa{at- tan.- 1 (a -/3)/h} 7 
2Wo {(a-j8) 2 + h 2 y da ’ 
and the energy contained between a and a -{-da is consequently 
1 da 
4 tt {(a - f3) 2 + h 2 }’ 
Now h is supposed to be small in comparison with f3. Consequently the 
above expression has a sharp maximum falling away to zero on both sides. 
If the train represented by e~ ht cos /3t from t = 0 to t — oc is received by the 
same ideal spectroscope, a line of finite breadth is formed at the point in 
* Phil. May., xxvii, pp. 460-469, 1889 ; or Collected Works , vol. iii, p. 268. 
