Study oj Huygens's Secondary Waves. 209 



inside the plane of the aperture. But Kirchhoff's formula 

 leads to a different result. An element of the surface AB at 

 X would have an effect proportional to 



(sin |PxB -sin lDxA ), 



and if the angle PAB be not very large, the path-difference 

 PA 4- AD — PB — BD would not be very large compared with 

 X and therefore there ought to be a finite effect at D. This 

 absurd result discredits the applicability of Kirchhoff's formula 

 to experiment, and further shows that the investigation of 

 the correct obliquity factor has an actual tangible relation to 

 experiment. 



We now proceed to obtain a solution of the general 

 differential equation of wave-motion in the form of a surface- 

 integral which satisfies the requisite boundary conditions. 

 Let S be the surface over which the integration is to be 

 effected and r be the distance of a point in the space around 

 from an element <2S of the surface. 



r 



is a solution of the symbolical equation 

 (V 2 + P)=0. 

 If dn be an element of the normal to the element dS, then 



d /e~ lkr \ 

 dn\ r ) 



is also a solution of the differential equation. If <f> is an 

 expression which is a function of the position of dS on the 

 surface but does not contain r, then 



Mfi(^> s w 



is a solution of the equation. This integral is the well-known 

 expression for the potential of a sheet of double sources of 

 sound, provided one of the directions of the normal n be 

 regarded as positive and the other negative. The value of 

 the integral at the surface itself is, on one side of it + <£, on 

 the other —<j> : for, regarding one of the directions of the 

 normal as positive, the value of the integral at a point 

 Phil. Mag. S. 6. Vol. 17. No. 97. Jan. 1909. P 



