ourable circumstances only 1/m^2[pi]^2 of the original
light can be obtained in the m^th spectrum. We conclude that, with a
grating composed of transparent and opaque parts, the utmost light
obtainable in any one spectrum is in the first, and there amounts to
1/[pi]^2, or about 1/10, and that for this purpose a and d must be
equal. When d = a the general formula becomes

sin^2 1/2m[pi]
Bm : B = ------------- (3),
m^2[pi]^2

showing that, when m is even, Bm vanishes, and that, when m is odd,

Bm : B = 1/m^2[pi]^2.

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 1 2 / 1 1 \
1 = - + - + ------ ( 1 + - + -- + ... ),
2 4 [pi]^2 \ 9 25 /

which is true by a known theorem. In the general case

___m=[oo]
a / a \^2 2 \ 1 /m[pi]a\
----- = ( ----- ) + ------ > -- sin^2( ------ ),
a + d \a + d/ [pi]^2 /__ m^2 \ a + d/
m=1

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).

[Illustration: FIG. 6.]

From the value of Bm : B0 we see that no lateral spectrum can surpass
the central image in brightness; but this result depends upon the
hypothesis that the ruling acts by opacity, which is generally very
far from being the case in practice. In an engraved glass grating
there is no opaque mater