Of the Rays of Light. 173 
Now if aB be denoted by p, Bo by d, and om by 2, it is obvious that 
e=P+(p—-ryY, S=ae+(p+ax). 
T’rom these equations we have approximately 
es p—r\?* i ‘aa\el a 
aad 4i+4( = Jet sadfi+3( sty se 
oS = =2 tana a, 
and therefore 
denotipg the angle aos by a. Hence the general expression of the intensity’of the light, 
at any point M, is 
A’ =a? + 2ad' cos (4 tana x) +a?. 
Again, substituting for & —8 its value just found, we see that the successive fringes 
will be formed at the distances given by the formula 
x= md cotan a; 
in which m is any number of the natural series, its even values giving the places 
of the bright fringes, and its odd values those of the dark ones. Accordingly, the 
bright fringes are formed at the distances 0, 2/, 4/, &c., and the dark ones at the 
distances intermediate, /, 3/, 51, &c., 1 being equal to +  cotan a: the successive 
fringes, therefore, are equidistant. It is obvious that the angle a must be very small, 
or the incidence very oblique, in order that the fringes should have any sensible breadth. 
We have hitherto assumed that the light has undergone no change by reflexion, ex- 
cepting the change of direction. Let us now suppose that the phase of the vibration 
is accelerated, and let us examine the effect produced in the position of the fringes. 
Let the amount of this acceleration be denoted by the angle pz ; then the difference 
of the phases*will be 
s—8 cm &—8—Lyr 
2n(2—*) = pr = 2x (—*). 
So that the successive fringes will be formed at the points for which 
0 —s—$prA = km), 
m being any number of the natural series. But we have already found that 
&—8§=2 tan a «; so that the points in question are given by the formula, 
a=} (m+) d cotan a; 
the even values of m corresponding to the bright fringes, and the odd values to the 
dark ones. It is evident from this that the magnitude of the fringes will be unal- 
