Cambridge Philosophical Society, 237 



P as the resultant of the elementary disturbances corresponding to 

 the several elements of P. Let it be required to determine the dis- 

 turbance which corresponds to an elementary portion only of the 

 plane P. In this consists the whole of the dynamical part of the 

 theory of diffraction, if we except the case of diffraction at the com- 

 mon surface of two different media ; the rest is a mere question of 

 integration. Let the time t be divided into equal intervals, each 

 equal to r. The disturbance which is propagated across the plane 

 P during the first interval r occupies a layer of the medium having 

 a thickness vr, if v be the velocity of propagation, and consists of a 

 certain velocity and a certain displacement. By the problem above 

 mentioned, we can find by itself the effect of the disturbance which 

 occupies so much only of this layer as corresponds to a given element 

 rfS of P. By doing the same for the 2nd, 3rd, &c. intervals r, and 

 then making the number of such intervals increase and their magni- 

 tude decrease indefinitely, we shall get the effect of the disturbance 

 which is continually transmitted across (/S. The result is a little 

 complicated, but is much simplified when certain terms are neg- 

 lected which are only sensible when the radius of the secondary 

 wave is comparable with K, and which are wholly insensible in the 

 physical applications of the problem. The result thus simplified 

 may be enunciated as follows : — In the enunciation, the term 

 diffracted ray is used to denote the disturbance in an elementary 

 portion of a secondary wave, diverging in a given direction from the 

 centre ; the plane containing the incident and diffracted rays will 

 be called the plane of diffraction, the supplement of the angle be- 

 tween these two rays the angle of diffraction, and the plane passing 

 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 



?=csin — (vt—x) 



be the displacement in the case of the incident light, X! 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 Z, 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 f with the direction of vibration ; 

 then 



5'=:£l^(l + cos9)sin^cos?^(t;f-r) . . . (a.) 

 2X' A 



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

 not f(vt—i') hntf(vt—r) that appears in t', where/' denotes the 

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



