BANDS SEEN IN THE SPECTRUM, 
235 
the effect of interference has been, not to annihilate any light, but only to alter the 
“distribution of the illumination,” so that the light, instead of being diffused over 
the screen, is concentrated in front of the aperture. 
Now in the case of the bands considered in Section I., if we suppose the plate 
extremely thin, the bands will be very broad ; and the displacement of illumination 
due to the retardation being small compared with the breadth of a band, it is evident, 
without calculation, that at most only faint bands can be formed. This particular 
example is sufficient to show the inadequacy of the imperfect theory, and the neces- 
sity of an exact investigation. 
11. Suppose first that a point of homogeneous light is viewed through a telescope. 
Suppose the object-glass limited by a screen in which there is formed a rectangular 
aperture of length 2 /. Suppose a portion of the incident light retarded, by passing 
through a plate bounded by parallel surfaces, and having its edge parallel to the 
length of the aperture. Suppose the unretarded stream to occupy a breadth h of the 
aperture at one side, the retarded stream to occupy a breadth k at the other, while 
an interval of breadth 2g exists between the streams. In the apparatus mentioned 
in Section I., the object-glass is not limited by a screen, but the interfering streams 
of light are limited by the dimensions of the fluid prism, which comes to the same 
thing. The object of supposing an interval to exist between the interfering streams, 
is to examine the effect of the gap which exists between the streams when the retard- 
ing plate is inclined. In the investigation the effect of diffraction before the light 
reaches the object-glass of the telescope is neglected. 
Let O be the image of the luminous point, as determined by geometrical optics,/" 
the focal length of the object-glass, or rather the distance of O from the object-glass, 
which will be a little greater than the focal length when the luminous point is not 
very distant. Let C be a point in the object-glass, situated in the middle of the 
interval between the two streams, and let the intensity be required at a point M, 
near O, situated in a plane passing through O and perpendicular to OC. The in- 
tensity at any point of this plane will of course be sensibly the same as if the plane 
were drawn perpendicular to the axis of the telescope instead of being perpendicular 
to OC. Take OC for the axis of z, the axes of x and y being situated in the plane 
just mentioned, and that of y being parallel to the length of the aperture. Let p, q 
be the coordinates of M ; x,y, z those of a point P in the front of a wave which has 
just passed through the object-glass, and which forms part of a sphere with O for its 
centre. Let c be the coefficient of vibration at the distance of the object-glass ; then 
we may take 
("•) 
to represent the disturbance at M due to the element dxdy of the aperture at P, P 
being supposed to be situated in the unretarded stream, which will be supposed to 
lie at the negative side of the axis of x. In the expression {a.) it is assumed that the 
2 I 
MDCCCXLVIII. 
