262 Lord Rayleigh on the Passage of Waves 



major axis a and eccentricity e. We have 



™=WY (15) 



F being the symbol of the complete elliptic function of the 

 first kind. When e = 0, ¥(e)=^7r ; so that for a circle 

 M = 2a/7r. 



It should be remarked that \P in (9) is closely connected 

 with the normal velocity at dS. In general, 



£-JK<S> <«' 



At a point (■#) infinitely close to the surface, the neigh- 

 bouring elements only contribute to the integral, and the 

 factor e~ ikr may be omitted. Thus 



dx 



= -jJV^S=-2™£ ¥=? = -»»*■ 



or 



*=-«?#, (17) 



dyfr/dn being the normal velocity at the point of the surface 

 in question. 



Boundary Condition <f> = 0. 



We will now suppose that the condition to be satisfied on 

 the walls is </) = 0, although this case has no simple applica- 

 tion to aerial vibrations. Using a similar notation to that 

 previously employed, we have as the expression for the 

 principal solution 



Xm=e -**- e ito Xp=0y . . . (18) 



giving over the whole plane (x=0), % TO = 0, % P =0, 

 dxm/dx — — 2 ik, dx P /dx = 0. 



The supplementary solutions now take the form 



♦.-JJiF^-* MJiK>- JS - (1 " 



These give on the walls ^r m = ^r p = 0, and so do not disturb the 

 condition of evanescence already satisfied by %. It remains 

 to satisfy over the aperture 



*■.=+,> -M + d+Jdx=d+Jdx. . . (20) 



The first of these is satisfied if ty m = — ^ p) so that yjr m and 

 TJr P are equal at any pair of corresponding points upon the 

 two sides. The values of dty m /dx, d^ p /dx are then opposite, 



