6 ; REPORT—1848, 
face, and directed into the first medium, both being measured parallel to the plane 
of incidence; and let the expression for the vibration be put under the form 
@sin=* (sini. #4 cosi.y + 0d, bsin — (sini. e—cosi.y + v4), or 
csin 22(5 (sin 2’. « + cos2!.y] + vt), 
aceording as the incident, reflected or refracted vibration is considered. Whenever 
the coefficient of vibration becomes imaginary, put it under the form esi, retain 
the modulus ¢ for the coefficient, and subtract 6 from the phase. Whenever the coefti- 
cient of y under the circular function becomes imaginary, and equal to + k ./—j, 
remove y from the circular function and multiply by the exponential sy 
This rule having been established, the calculation of the intensity presents no 
difficulty. If I be the intensity of the reflected light, that of the incident light being 
unity, it is found that 
v4 2 
Be oe NE 
(€—qg)?+ 4qsin?26 
In this expression it is supposed that the two media between which the spot is 
formed are of the same kind, and that the incident light is polarized, either in the 
plane of incidence, or in a’ plane perpendicular to the plane of incidence: g is the 
12T / sini —1 
same in the two cases, being equaltos % » where T is the thickness 
of the plate of air at the point considered ; but 6 is different, being equal to @, in the 
former case, and 6, in the latter, where 
p tan 6, = tan b= sccig / i sin? i 
& 
When the vibrations take place in the plane of incidence, it would be necessary to 
consider separately the resolved parts of the vibration parallel to w and parallel to y; 
but the same expression would have been obtained for I if this consideration had 
been neglected. It is unnecessary to write down the expression for the intensity of 
the transmitted light, since the sum of the two intensities is equal to unity. For 
this reason it will be sufficient to discuss the intensity of the reflected light. 
From the expression for I the author has deduced the following consequences : 
1. At the point of contact T= 0; and on receding from that point T varies as 7°, 
ry being the radius vector measured from the point of contact. Hence at the point 
of contact there is absolute darkness; on receding from that point the intensity in- 
creases, at first very slowly, varying ultimately as 7*, so that for some distance round 
the centre the darkness is as to sense perfect; then the intensity increases more 
rapidly, and then it very rapidly approaches its limiting value 1. This agrees with 
observation. \ 
2. For different colours, the same fraction of the incident light is reflected at 
points for which r variesas 4/ a. Hence the spot is larger for red light than for violet ; 
but the separation of colours is small. This agrees with observation. 
3. When the angle of incidence lies between the critical angle and the angle 
1 
sin! ( ; + )' the spot is larger for light polarized in a plane perpendicular to 
1+ 
the plane of incidence than for light polarized in the plane of incidence; while be- 
tween the latter angle and 90° the reverse is the case. This difference of size agrees 
with observation, but it is impossible to say at what angle the change takes place. 
4. Suppose the internally incident light polarized at an azimuth of 45°, or there- 
abouts; and let the transmitted light be analysed so as to darken the centre of the 
spot; then a faint ring of light ought to be seen separating the dark centre from the 
generally dark field of view. This ring ought to be slightly bluish inside and reddish 
or brownish outside. The author has not tried this experiment. 
iB ee n 
