AN OBJECT-GLASS WITH CIRCULAR APERTURE. 289 



I shall now apply the numbers of the table to the solution of 

 the following problem. To find the diameters, &c. of the rings when 

 a circular patch, whose diameter is half the diameter of the object- 

 glass, is applied to its center, so as to leave an annular aperture. 



The radius of the patch being -, it is easily seen that the dis- 

 placement (using the same notation) is 



2sm-—-(vt—J'—A)fr\/a^-x'.cos—-.^a; (from a;—-a to x=+a) 

 - 2sin ~(vt-f-A)J\/---af. cos-^ .^x (from x= -- to x= +^. 

 Putting - =w, — = u, this becomes 



4a^ . sin -T-{vt —f — A) /„ \/l — vf . cos — .—^w 



A A / 



-4.^.sm yC^^-/- ^)/«vl-M'.cos— .— .M, 



the limits of integration both for w and for u being and 1. Omitting 

 the factor oV, the intensity will be expressed by 



V(»)-i*(l)}". 



where (p{n) is the number given in the table. 



Upon forming the numerical values we find that the black rings 

 correspond to values of w=3,15, 7,18, 10,97: and that the intensities 



of the bright rings (in terms of the intensity of the center) are — , — . 



Thus the magnitade of the central spot is diminished, and the bright- 

 ness of the rings increased, by covering the central part of the object- 

 glass. 



In like manner, if the diameter of the circular patch = a ( 1 — />), the 

 intensity of light would be proportional to {<p {n) — {l— pf .^{n—pn)}". 



