very nearly
hit the mark.[3] And a little before Helmholtz, E. Abbe published a
somewhat more complete investigation, also founded upon the phenomena
presented by gratings. But although the argument from gratings is
instructive and convenient in some respects, its use has tended to
obscure the essential unity of the principle of the limit of resolution
whether applied to telescopes or microscopes.

[Illustration: Fig. 4.]

In fig. 4, AB represents the axis of an optical instrument (telescope
or microscope), A being a point of the object and B a point of the
image. By the operation of the object-glass LL' all the rays issuing
from A arrive in the same phase at B. Thus if A be self-luminous, the
illumination is a maximum at B, where all the secondary waves agree in
phase. B is in fact the centre of the diffraction disk which
constitutes the image of A. At neighbouring points the illumination is
less, in consequence of the discrepancies of phase which there enter.
In like manner if we take a neighbouring point P, also self-luminous,
in the plane of the object, the waves which issue from it will arrive
at B with phases no longer absolutely concordant, and the discrepancy
of phase will increase as the interval AP increases. When the
interval is very small the discrepancy, though mathematically
existent, produces no practical effect; and the illumination at B due
to P is as important as that due to A, the intensities of the two
luminous sources being supposed equal. Under these conditions it is
clear that A and P are not separated in the image. The question is to
what amount must the distance AP be increased in order that the
difference of situation may make itself felt in the image. This is
necessarily a question of degree; but it does not require detailed
calculations in order to show that the discrepancy first becomes
conspicuous when the phases corresponding to the various secondary
waves which travel from P to B range over a complete period. The
illumination at B due to P then becomes comparatively small, indeed
for some forms of aperture evanescent. The extreme discrepancy is that
between the waves which travel through the outermost parts of the
object-glass at L and L'; so that if we adopt the above standard of
resolution, the question is where must P be situated in order that the
relative retardation of the rays PL and PL' may on their arrival at B
amount