204 On the Manufacture and Theory of Diffraction-gratings. 



The result is 



C™ dz 2tt, A/ x/\ 2tt/ A/ -„ \\ 



showing that the total effect is retarded - behind that due to the 



o 



element at 0. This result is analogous to, though different from, 

 that of the ordinary integration by Huyghens's zones. In that 

 case the effect of each zone is very nearly the same, and therefore 

 the whole is the half of that of the first zone. If the first zone be 

 divided into rings by circles drawn so that r increases in arith- 

 metical progression, the rings will be of equal area, and there- 

 fore the phase of the resultant vibration will be halfway between 

 that corresponding to the centre and circumference — that is, 

 will be retarded relatively to the centre by one fourth of X. In 

 the present case, if the bar be divided on the same principle so 



that each piece gives a result retarded ~ behind its predecessor, 



the lengths will rapidly diminish from the centre outwards, and 

 therefore the same argument does not apply. The retardation 

 of the resultant relatively to the central element is less than 

 before, on account of the preponderance of the central parts. 



By the result investigated in my paper previously referred to, 

 if T be the volume of the element P Q, D and D' the original 

 and altered densities, the disturbance at A due to the element is 



D'-Dtt.T . 2w /JL , x - 



___ rm «co. x (tt-r)*. ^ 



2tt 

 the original vibration at P Q being denoted by cos — bt. « is 



the angle between the ray PA and the direction of original vi- 

 bration Z ; but in the present application we may put sin a = 1, 

 since only the central parts are really operative. If we replace 

 T by Adz, A being the sectional area of the bar, and use the in- 

 tegral just investigated, we find for the effect of the whole bar 



D'-D Att /\ 2tt/. _ X\ 



2-7T 



corresponding to cos -r- bt for the incident light. 



A. 



When the original vibration is parallel to y, the disturbance 

 due to the bar will no longer be symmetrical round Z. If a 

 be the angle between x and the direction of the scattered ray, 

 it is only necessary to introduce the factor cos u in order to 

 make the preceding expression applicable. 



* The factor tt was inadvertently omitted in the original memoir. 



