87 



through a ray of plane-polarized light and containing the direction 

 of vibration the plane of vibration. 



The incident ray being plane-polarized, each diffracted ray will 

 be plane-polarized, and the plane of polarization will be determined 

 by the following law : — The plane of vibration of the diffracted ray is 

 parallel to the direction of vibration of the incident ray. The direction 

 of vibration being thus determined, it remains only to specify its 

 magnitude. Let 



C=esin — (vt—x) 

 A 



be the displacement in the case of the incident light, £' the displace- 

 ment in the case of the diffracted ray, £' being reckoned positive in 

 the direction which makes an acute angle with that in which £ is 

 reckoned positive. Let r be the radius of the secondary wave diver- 

 ging from dS, and let r make angles 9 with the direction of propa- 

 gation of the incident ray, and <p with the direction of vibration ; 

 then 



?'= — (1 -+- cos 9) sin <p cos — (vt—r) . . . (a.) 

 Z\r a 



When an arbitrary function of vt — x, f(vt — so) occurs in £, it is 

 not f (vt—r) but f ''(vt—r) that appears in £', where/' denotes the 

 derivative of f, and accordingly in the particular case in which 

 f(u) = sin u the sine in £ is replaced in £' by a cosine. It may readily 

 be verified, that if the formula (a.) be applied to determine by inte- 

 gration the disturbance which corresponds to the whole of the plane 

 P, the disturbance in front is the same as if the wave had not been 

 supposed broken up, and no disturbance is propagated backwards. 



The law obtained for determining the position of the plane of po- 

 larization of the diffracted ray seems to lead to a crucial experiment 

 for deciding between the two rival theories between the directions of 

 vibration in plane-polarized light. Suppose the incident light po- 

 larized by transmission through a Nicol's prism mounted in a gra- 

 duated instrument, and let the diffracted light be analysed in a similar 

 manner. By means of the graduation of the polarizer, we can turn 

 the plane of polarization of the incident ray, and consequently the 

 plane of vibration, which is either parallel or perpendicular to the 

 plane of polarization, round through equal angles of say 5° or 10° 

 at a time. According to theory, the planes of vibration of the dif- 

 fracted ray will not be distributed uniformly, but will be crowded 

 towards the perpendicular to the plane of diffraction. But experi- 

 ment will enable us to decide whether the planes of polarization are 

 crowded towards the plane of diffraction or towards the perpendicular 

 to the plane of diffraction, and we shall accordingly be led to con- 

 clude, either that the vibrations are perpendicular, or that they are 

 parallel to the plane of polarization. 



In ordinary cases of diffraction, the illumination, in consequence 

 of interference, is insensible beyond a small angle of diffraction. It 

 is only by means of a fine grating that we can procure light of con- 

 siderable intensity that has been diffracted at a large angle. The 



