ON OPTICAL THEORIES. 203. 
in the incident and diffracted rays make angles a,, a, with the normal to 
the plane of diffraction. Then the diffracted ray and the two directions 
of vibrations lie in the same plane, and the directions of vibrations are 
normal to the respective rays. Thus, if we form a spherical triangle by 
drawing lines from the centre of a sphere, parallel to the normal to the 
plane of diffraction and to the two directions of vibrations, since the 
direction of vibration in the diffracted wave is the projection on that 
wave of the direction of vibration in the incident wave, we have 
cos9=tana,cota; . : : . (83) 
Now, let a and a be the azimuths of the planes of polarisation of the 
incident and diffracted light, measured from a plane normal to the plane 
of diffraction. Then, on Fresnel’s assumption that the direction of vibra- 
tion is normal to the plane of polarisation, we have 
wait, a= 5+ Cay 
and tan a =sec 6 tan a; 
while on MacCullagh’s hypothesis 
SG) SAGE 
and 
tana=cos@tanew . : , . (84) 
These two formule can be tested by experiment, and afford a means, 
therefore, of deciding between the two theories of reflexion, and of deter- 
mining the question whether reflexion be due to a change of density or 
to a change of rigidity in the ether; for the values of a corresponding to a 
series of values of 2 can be observed for any given angle of diffraction, 
and if the values of a be taken at equidistant intervals, the values of a, 
and therefore the positions of the plane of polarisation of the diffracted 
light, will not be equidistant, but will on the first hypothesis be crowded 
towards the plane of diffraction, while on the second they will be crowded 
away from that plane. 
Professor Stokes was the first to carry out a series of observations of 
this nature; he employed a grating ruled on glass at the rate of 1,300 
lines to the inch, and the results of his experiments are decisive in favour 
of Fresnel’s hypothesis. The experiments are troublesome, and the com- 
parison of the results with theory is complicated by the fact that the 
refraction through the glass plate on which the grating is ruled also 
produces a change in the position of the plane of polarisation. The 
amount of this change is the same on the two theories, and tends to 
produce a crowding of the planes of polarisation away from the plane of 
diffraction, an effect opposite to that produced by diffraction on Fresnel’s 
theory. Moreover, we may suppose that, when the ruled face of the grating 
is towards the incident light, either the diffraction takes place in air so 
that the wave enters the glass obliquely, or that the diffraction takes 
place in the glass after the light has entered the first surface normally, 
while when the ruled surface is away from the incident light the diffrac- 
tion may take place in air after passing normally through the glass, or in 
the glass so that the light after passing normally through the first sur- 
face emerges obliquely. 
