APRIL 25, 1884.] 
LETTERS TO THE EDITOR. 
Fy Correspondents are requested to be as brief as possible. 
The writers name is in all cases required as proof of good faith. 
‘A singular optical phenomenon.’ 
THE ‘singular optical phenomenon’ described by 
‘F. J. S.’ on p. 275 of the current volume of Science 
is a case of the familiar watering effect produced by 
superposed loose and regular fabrics, or by distant 
palings and lattice-works superposed by projection. 
We may find it convenient, in the following discus- 
sion, to refer to these by the general term of ‘ pro- 
jection phenomena,’ although the phrase does not 
seem to me to have much to recommend it except 
convenience. 
I ought to say that this discussion is prompted by 
the letter by Professor LeConte in the last number 
of Science ; for, if so skilled an experimenter could 
overlook the real explanation, it may safely be con- 
cluded that most readers have done so. Moreover, 
the phenomenon is one of a large and interesting 
class, of which I have never met any explanation, 
although, as we shall see, very simple considerations 
ae lead us far towards a complete explanation of 
all. 
For the sake of simplicity, we will begin by the 
consideration of two gratings of regular “horizontal 
elements: the one nearer the “observer, which we will 
call the first grating, is to be of alternating opaque 
and transparent strips; and the more distant one, or 
second grating, of white and black bands. We will 
also suppose, at first, that the eye is placed in a line 
passing through the middle of a dark band and an 
opaque strip, and that the aperture of the pupil is 
negligibly small. We may also conveniently assume 
that the angular widths of the elements of both 
gratings are so small that they are not separately 
evident to the eye, not only because such cases offer 
the most striking phenomena, but also because in 
them the meaning of the term ‘apparent DEBS 
which we shall use, is self-evident. 
We will call the distances from the eye to ane 
screens respectively d, and d,; the breadths of the 
opaque and black intervals, 6; and 02; and, finally, 
the element of each erating (that is, the distance 
from the centre of one dark strip to the centre of the 
next), Z, and EE). 
If B is the brightness of the white portion of the 
second grating, it is evident that the average bright- 
ness of the field, if the first grating were removed, 
would be 
Es ae De 
£2 
‘Tf, on the other hand, the first screen remained in 
place, and the black strips of the second should be 
replaced by white of brightness B, the field would 
appear of a brightness 
(peas 
Ly, 
As a first special case, let us suppose 
By Hz, 
B 
then, remembering the position of the eye, it is 
clear that each opaque bar would be centrally pro- 
jected upon a dark strip of the second grating; and 
the brightness would be uniform, and equal to the 
less of the two expressions above. 
For a second case, suppose 
SCIENCE. 
d01 
n being any whole number: then every nth black 
strip would be centrally covered by a bar of the first 
grating. If a is equal to or less than “2 the 
i 
“2 
e 
5, —h, 
brightness would be uniform, and equal to B =e ; 
but, if this limit of equality were surpassed, the 
average brightness would be 
nEy —(n—1) bg —b,2 
‘dy 
B NE, Z 
and there would be regularly placed minima, unless 
nie were insensible to the eye. 
ZZ 
E, Ey 
The case of n—* = —~ is equally easy. 
the angle -—— 
In all that follows, we will, in order to avoid too 
extensive discussion, regard n as equal to unity: by 
this limitation we sacrifice no interesting cases. 
Suppose, now, the eye moved continuously up or 
down, parallel to the gratings. After a certain small 
displacement, depending upon the relation of a to - 
1 2 
the brightness of the field would continuously dimin- 
ish until it reached a minimum equal to 
i, ¢ 
unless the numerator should be negative, when the 
minimum would be absolute. It would remain at 
this minimum for a certain time, depending upon 
the constants of the system, and then increase by ex- 
actly the same law as that of decrease, until after a 
displacement of the eye equal to Ds - when it 
would recur to the same condition as at first. 
As a final and more general case, let us suppose 
that 
where dis a small quantity, positive or negative. If 
we again suppose that the eye is so placed that a line 
drawn from it perpendicularly to the two gratings 
will pass centrally through dark bars in each, then a 
line drawn from the eye through the mth bar of 
the first grating will pass through a dark strip of the 
second, if = is a whole number. Let m be the 
2 
smallest number which meets this condition: then a 
line drawn through any bar between the Ist and mth 
would meet some one of the conditions discussed in 
the last paragraph, as produced by a movement of 
the eye. Thus we see that the field would present 
horizontal maxima and minima of brightness, the 
angular position (@) of the maxima being given by 
the equation 
mE, 
= tang) .N —= 
where WN is any whole number, positive or negative. 
The apparent distance apart of the maxima would 
be m Ey 
] 
If the eye be moved so as to shift the apparent 
position of the central bar to the adjacent black strip 
on the second grating, the middle of the field would 
have undergone all the changes of phase which cor- 
respond to a change of tang @ from zero to m=, 
J 
hence such a motion of the eye would appear to give 
