resolving power with aperture will
now be evident. The larger the aperture the smaller are the angles
through which it is necessary to deviate from the principal direction
in order to bring in specified discrepancies of phase--the more
concentrated is the image.

In many cases the subject of examination is a luminous line of uniform
intensity, the various points of which are to be treated as
independent sources of light. If the image of the line be [xi] = 0,
the intensity at any point [xi], [eta] of the diffraction pattern may
be represented by

[pi]a[xi]
_+[oo] sin^2---------
/ a^2b [lambda]f
| I^2d[eta] = --------- ------------- (8),
_/ [lambda]f [pi]^2a^2[xi]^2
-[oo] ---------------
[lambda]^2f^2

the same law as obtains for a luminous point when horizontal
directions are alone considered. The definition of a fine vertical
line, and consequently the resolving power for contiguous vertical
lines, is thus _independent of the vertical aperture of the
instrument_, a law of great importance in the theory of the
spectroscope.

The distribution of illumination in the image of a luminous line is
shown by the curve ABC (fig. 3), representing the value of the
function sin^2u/u^2 from u = 0 to u = 2[pi]. The part corresponding to
negative values of u is similar, OA being a line of symmetry.

[Illustration: Fig. 3.]

Let us now consider the distribution of brightness in the image of a
double line whose components are of equal strength, and at such an
angular interval that the central line in the image of one coincides
with the first zero of brightness in the image of the other. In fig. 3
the curve of brightness for one component is ABC, and for the other
OA'C'; and the curve representing half the combined brightnesses is
E'BE. The brightness (corresponding to B) midway between the two
central points AA' is .8106 of the brightness at the central points
themselves. We may consider this to be about the limit of closeness at
which there could be any decided appearance of resolution, though
doubtless an observer accustomed to his instrument would recognize the
duplicity with certainty. The obliquity, corresponding to u = [pi], is
such that the phases of the secondary waves range over a complete