aperture.

If, on the other hand, we suppose the aperture given, we find that
aberration begins to be distinctly mischievous when it amounts to
about a quarter period, i.e. when the wave-surface deviates at each
end by a quarter wave-length from the true plane.

As an application of this result, let us investigate what amount of
temperature disturbance in the tube of a telescope may be expected to
impair definition. According to J. B. Biot and F. J. D. Arago, the
index [mu] for air at t deg. C. and at atmospheric pressure is given by

.00029
[mu] - 1 = -----------.
1 + .0037 t

If we take 0 deg. C. as standard temperature,

[delta][mu] = -1.1 X 10^-6.

Thus, on the supposition that the irregularity of temperature t
extends through a length l, and produces an acceleration of a quarter
of a wave-length,

1/4[lambda] = 1.1 lt X 10^-6;

or, if we take [lambda] = 5.3 X 10^-5,

lt = 12,

the unit of length being the centimetre.

We may infer that, in the case of a telescope tube 12 cm. long, a
stratum of air heated 1 deg. C. lying along the top of the tube, and
occupying a moderate fraction of the whole volume, would produce a not
insensible effect. If the change of temperature progressed uniformly
from one side to the other, the result would be a lateral displacement
of the image without loss of definition; but in general both effects
would be observable. In longer tubes a similar disturbance would be
caused by a proportionally less difference of temperature. S. P.
Langley has proposed to obviate such ill-effects by stirring the air
included within a telescope tube. It has long been known that the
definition of a carbon bisulphide prism may be much improved by a
vigorous shaking.

We will now consider the application of the principle to the formation
of images, unassisted by reflection or refraction (_Phil. Mag._,
1881). The function of a lens in forming an image is to compensate by
its variable thickness the differences of phase which would otherwise
exist between secondary waves arriving at the focal point from various
parts of the aperture. If we suppose the diameter of the lens to be
given (2R), and its focal length f gradually to increase, the original
differences of phase at the image of an infinitely distant luminous
point diminish without limit. When f attains a certain value, say f1,