AN OBJECT-GLASS WITH CIRCULAR APERTURE. 285 



The first integration with regard to y is simple, as y does not 

 enter into the expression, which is therefore to be considered as con- 

 stant. Putting y, and y^ for the smallest and greatest values of y 

 corresponding to x, the first integral is 



{yi-yx)y-^m-^{vt-f-A^r-x). 



To this point of the investigation the expressions are general, including 

 every form of contour of the object-glass. 



We must now substitute the values of y^ and y^ in terms of x, 

 before integrating with regard to x. For a circular aperture 



y, - y. — ^y/a^-x" 



where the sign of the radical is essentially positive. Hence the dis- 

 placement of the ether at the point defined .by the distance A is re- 

 presented by 



2 f, Va' - x" . sin — {vt-f- A + ^x) 

 = 2sm -^{vt-/— A) f^\/a^-af .cos-— .^x 



\ Ay 



+ 2cos — - {vt —f— A) X a/«^ — x\sm—-.^x, 



A ^ J 



and the limits of integration are from x = — a to x = + a. Between 

 these limits it is evident that 



;- /-: « . 2-ir b 



f^Va' — x^ .sm—- . ^x = 0, 



^ J 



(as every positive value is destroyed by an equal negative value) ; and 

 the displacement is therefore represented by 



2sin— -(«^— /— ^) ji\/«^ — ar'.cos ^ .^x, 

 ^ ■ , ^ ./ 



the integral being taken between the limits x= -a, x— -^a. 



p p2 



