The third spectrum has thus only 1⁄9 of the brilliancy of the first.
Another particular case of interest is obtained by supposing a small relatively to (a + d). Unless the spectrum be of very high order, we have simply
Bm : B = {a/(a + d)}² (4);
so that the brightnesses of all the spectra are the same.
The light stopped by the opaque parts of the grating, together with that distributed in the central image and lateral spectra, ought to make up the brightness that would be found in the central image, were all the apertures transparent. Thus, if a = d, we should have
1 = ½ + ¼ + 2/π² (1 + 1⁄9 + 1⁄25 + ...),
which is true by a known theorem. In the general case
| a | = ( | a | ) | ² | + | 2 | Σ | m=∞ | 1 | sin²( | mπa | ), |
| a + d | a + d | π² | m=1 | m² | a + d |
a formula which may be verified by Fourier’s theorem.
According to a general principle formulated by J. Babinet, the brightness of a lateral spectrum is not affected by an interchange of the transparent and opaque parts of the grating. The vibrations corresponding to the two parts are precisely antagonistic, since if both were operative the resultant would be zero. So far as the application to gratings is concerned, the same conclusion may be derived from (2).