172 On a New Case of Interference 
stances it is somewhat strange that the fact of the interference of direct and reflected 
lights should not have been itself submitted to the test of experiment ; especially as 
the character of this interference, if it were found to exist, might be expected to throw 
some light upon the laws of reflexion itself. 
The theory of such interference is easily deduced from the general principles. Let. 
light proceeding from a single luminous origin fall upon a reflecting surface, at an 
incidence of nearly 90°: a screen placed at the other side of the reflector will be 
illuminated, throughout a certain extent, by both direct and reflected lights; and, if 
the difference of the paths traversed by these lights amount only to a small multiple 
of the length of an undulation, the two lights will form fringes by their interference. 
Let the intensities of the direct and reflected lights be denoted by a’ and a*, and 
that of the resulting light by 4’; then, by the theory of the composition of coexisting 
vibrations, we have 
5 r o—8 
J=7> 2 I, 
A=a +2 aa’ cos Ix ( x ) +a : 
3 and 8 denoting the lengths of the paths traversed by the two waves, from their origin 
to any given point, and ) the length of an undulation. 
The intensity of the resulting light will be a maximum, and equal to (a +a’), at 
those points for which 
cos 2m (2—*) = + as or 8—d= ans ; 
It will be a minimum, and equal to (a—da' ), when 
cos 2x (2) = 1, or 8—8= (@n+19 
D > 
n being any number of the natural series 0, 1, 2, 38, &e. Bright fringes, therefore, 
will be formed at all the points included in the former equation, and dark ones at the 
points corresponding to the latter. 
i 
A 
Let op be the reflector, om the screen placed in contact with it, and perpen- 
dicular to its plane; and let a be the luminous origin, and a’ its reflected image at 
an equal distance below the line os ; then if m be any point, whose illumination is 
required, 8=am, 6 =a'M. 
