268
Dioptries of the Eye
[231, 232. G.
ished until the line in question becomes so short that it is indistinguish¬
able from a piece of any other curved line. The general constitution
of a bundle of optical rays may be defined also by the fact that the
two focal lines with respect to any given ray are intersected by all the
contiguous rays (disregarding infinitesimal aberrations of order higher
than the first); but this statement requires an explicit definition of
what is meant by a contiguous ray. Draw the ray corresponding to a
point on the ruled surface at a finite distance from the given ray,
and do the same thing for points nearer and nearer the given ray ; then
ultimately the distance between the point and the given ray will
vanish entirely, and the ray corresponding to this point will be con¬
tiguous to the given ray. But this definition will be fundamentally
misunderstood if the conclusion is deduced that all the rays of an
infinitely narrow bundle go approximately through two Sturm focal
lines.
Since (neglecting infinitesimals of order higher than the first) all
the contiguous rays of a bundle of rays which is anastigmatic along a
certain ray meet this ray in the focal point, there occurs in this case a
perfect ray-convergence of the first order ; and the focal point is the optical
reproduction or image of the point where the light originates. Hence,
in case of a stop of finite aperture, the optical imagery cannot be said
to be due to the fact that all the rays coming from an object-point go
approximately through the image-point, but merely to the fact that
contiguous rays intersect at this point, so that the concentration of the
light here is infinitely great as compared with that at a point at a finite
distance from this spot. The imagery supposed in the first case above
is indeed mathematically correct for an infinitely narrow bundle of
rays, but since it is physically impossible on account of diffraction
effects to produce an optical image by such means, this imagery repre¬
sents merely an ideal dreamed of in olden times; whereas the latter
case mentioned above describes exactly the actual process as it takes
place. If the reality here appears to be little different from the ideal,
the explanation is to be found in the circumstance that both in the
eye and likewise on the photographic plate shades of brightness are of
more importance than absolute brightness.
The criterion of actual optical imagery is just this perfect ray-
convergence of the first order. As is evident from the foregoing, this
convergence of the rays is susceptible of mathematical investigation
only along definite rays. The rays chosen for this purpose are the chief
rays defined above. Assuming that the chief ray corresponding
to each point of the surface of the object has been constructed
and traced through the optical system by trigonometrical calcula¬
tion, and that somewhere in the path of the light a screen is