602 
rise to a shifting of the whole series of maxima by 
this angle. The direction of apparent motion would 
be either with that of the eye, or opposite, according 
as 0 is positive or negative. The displacement of 
the pupil necessary to bring about this change would 
be 
Pippa cual 
de roa dy 
If the relative motion of the periodic phenomenon 
and the first screen be regarded as a parallactic dis- 
placement, then we must suppose their relative dis- 
tances from the eye inversely proportional to their 
apparent motions; 1.e., as 
nH 4 
mE 4, Ey : 
d, dy ram dy 
ae E; 
or, since 2) nearly, as m to - de 
1 2 2 1 
It was this apparent parallax which led ‘F. J.S.’ to 
suppose the phenomenon which he describes an image 
of the distant screen between himself and the first 
window. 
If our gratings be complicated by the addition of 
equally spaced vertical bars, we shall see also, in gen- 
eral, a series of vertical bands giving maxima and 
minima along a horizontal direction. ‘These will be 
separated by intervals 
m EF’, 
Cin 
and the ratio of their apparent angular motion to 
that of the first screen when the eye is moved equals 
d. 
i Ow 
: 2 1 
where the letters marked with / are defined by anal- 
ogy. 
“ON very interesting conclusion follows from the 
consideration that mand m/ are wholly independent; 
the one depending on 0, and the other on 06’. Thus, 
we may have the horizontal bands moving in the 
same direction as the eye, and the vertical bands 
moving in the opposite direction, or vice versa: 
hence, if the displacement of the eye is neither 
horizontal nor vertical, the network which forms the 
projection phenomenon may seem to move in any 
direction, the only condition being that the horizontal 
and vertical components of the velocity are propor- 
tional, respectively, to m/ and m; or, in other words, 
to the apparent width of the bands, divided by the 
corresponding element of the first grating. 
In the case of gratings which are not plane, super- 
posed by projection, as is the condition generally 
with doubled laces, veils, mosquito-bars, ete., —in 
short, in almost all cases of every-day observation, — 
both 6 and 0’, as well as the direction of the elements 
of the gratings, are functions of the distances from 
the central point of the field; but, as these are con- 
tinuous functions, we can state several of the most 
important properties of the projection phenomena: 
viz., — 
1°, The bands will be continuous and curved. 2°. 
If the eye be moved, the phenomenon will shift with 
an apparent velocity in any direction proportional 
to the width of the bands measured in that direction. 
3°. The motion of a single band will, in general, be 
a motion of translation, combined with a motion of 
rotation. But the instantaneous centre of rotation 
cannot lie in a band; for in that case, according to 
the previous conclusion, that point being at rest, the © 
_ band would there have no width, consequently could 
not exist. 4°. If a band forms a closed curve, a 
- motion of the eye will necessarily produce a continu- 
ous change in the apparent magnitude of the ring; 
SCIENCE. 
for a mere motion of translation would correspond 
to a momentary rotation about at least two points in 
_ the curve, which, according to the last principle, is 
impossible. 
The properties described under the second and 
fourth heads above are those which more especially 
cause the projection phenomena to resemble those of 
watered silk; for the latter follow much the same law. 
We will now consider the effect of the size of the 
pupil of the observing eye, which has hitherto been 
considered as a point. It is obvious that the image 
on the retina must be the sum of the projection 
images as seen from each point of the pupil: hence, 
if the pupil is not much greater than the space 
through which the point of view must be shifted in 
order to produce a complete change of phase (i.e., 
than EF, a ), the phenomenon must be like that 
dz — dy 
for an indefinitely small pupil, except that the dis- 
continuity is less pronounced. This explains why, 
in fine networks, such as veils and mosquito-bars, the 
distance d, — d, between the fabrics must be small 
in order to produce the projection phenomena. In 
the case described by‘F. J. S.,’ Hz = 4 inch, d; = 10 
feet, and d, = 40 feet: consequently the expression 
indicating the limit which the diameter of the pupil 
must not greatly surpass is ¢ inch. 
The effect of maladjustment of the eye would be 
to diminish still further the discontinuity of the 
phenomenon; but this would be carried so far as to 
destroy the periodicity, and thus obliterate the phe- 
nomenon, — not when an angular interval of — 
2 
at the distance d, becomes indistinguishable, as 
‘F. J. 8.’ seems to have expected, but only when an 
mE, 
angular interval of d 
1 
at the distance d, becomes 
indistinguishable. 
The cases where n differs from unity offer no diffi- 
culties, but they are much less interesting. They 
exclude the case which has given rise to this discus- 
sion; for there £, equals 4 inch, the other dimensions 
having been already quoted. 
In what precedes, however, I have tacitly assumed 
that ae is always the reciprocal of a whole number. 
2 
This may not be true. 
tween L and 1 
N aN ot at 
then, if N is large, the solution above is accurate 
within the range of observation. If, on the contrary, 
the value of N is moderate, successive maxima will 
differ by a quantity which is itself periodic. 
It will be observed that the second grating may be 
perfectly replaced by an image by reflection of the 
first. Frequent examples of this arrangement are 
seen in screens before closed windows or mirrors. 
Suppose the value to lie be- 
, where W is a whole number: 
The general analytical solution of the whole class” 
of phenomena produced by parallel rectangular grat- 
ings with indefinitely small pupil is easy; but the 
solution is so extremely general, that its reduction to 
special interesting cases requires even more writing 
than we have found necessary here. The only point 
worth dwelling upon here is, that the apparent 
variations in brightness, though periodic, are always 
discontinuous; but that every departure from the 
assumed geometrical conditions, such as are effected 
by diffraction, dimension of the pupil, and imperfect — 
accommodation, tends to decrease the discontinuity. 
C. S. HASTINGS. 
Baltimore, April 11. : 
[Vou. IL, No.'64, 
