Theory of Diffraction- gratings. 203 



by small particles. Whether true or not, it is at any rate me- 

 chanically possible. 



Since the bars are very small, the effect of each is quite inde- 

 pendent of the rest ; and so the dynamical theory need only con- 

 cern itself with one. In my paper " On the Light from the 

 Sky "*, I proved that the effect of a body in disturbing the waves 

 of light incident upon it may be calculated by ordinary integra- 

 tion from those of its parts, provided that the square of the alte- 

 ration of mechanical properties may be neglected. This propo- 

 sition, though true as stated, requires some caution in use, and 

 is practically inapplicable when the body is elongated in the di- 

 rection of original propagation, because the dimension of the 

 body in this direction divided by X may occur as a factor in the 

 terms omitted. In the present case, however, where the light 

 is incident normally to the plane of the grating, this difficulty 

 does not arise. 



Let the bar under consideration be taken for axis of z, 

 and let the axis of x be 

 parallel to the direction of 

 propagation of the ori- J? 

 ginal light. The original 

 vibration is thus, accord- 

 ing to the polarization, 

 parallel to either z or y. 

 We will take first the 

 former case, where the 

 disturbance due to the bar 

 must be synfmetrical in all directions round O Z, and parallel to 

 it. The element of the disturbance at A due to P Q (dz) will 

 be proportional to dz in amplitude, and will be retarded in phase 

 by an amount corresponding to the distance r. In calculating 

 the effect of the whole bar, we have to consider the integral 



f-& %* lk , r ^ C0S ? (W ~ r) 



I — cos-— (bt— r) = \ - ^-^^ — . 



Jo r A, ; J B Vr 2 -K 2 



Now the denominator = v 7 ?* — RvV + R, a product of which the 

 second factor may be treated as constant in the integration in 

 view of the fact that the parts for which r differs much from R 

 destroy each other's effects. After this simplification the inte- 

 gral may be evaluated by means of the formula 



p sin£^ = p cosa^ At 



Jo ^ x Jo ^ x v 2' 



* Phil. Mag. February 18/1. 



