90 The Hon. J. W. Strutt on the Reflection of 



first ; and this cannot be done without the introduction of finite 

 surface- waves. Expressed in terms of <f> 3 yfr, they take the form 



(*+») Jf + (*-*) ^ +2n^= similar expression, 



"vsi + #" af / = similar expressiQn ; 



or on substitution of the values of </>, ty, with regard to (8), 

 <l>{m{a ,2 + b^-2nb 2 \ + 2nabY 



= cb l \m f {a^-}-b 2 )-2n>b 2 \+2n l a l bylr l , . . (9) 



n {bhfr, -a 2 X + ib(aX-a^ \ 



= n'{b c 'X-a i ^ l + ib(aY-a l ^ l )\. . . (10) 



Although «' 2 + Z> 2 is vanishingly small, we are not at liberty to 

 leave it out, because m{ a 12 + 5 2 ) is finite. In fact 



m(a' 2 +& 2 )=Dc 2 ,l Q1) 



m'(a/ 2 4-& 2 )=D'c 2 J ' " ^ ' 



for we may neglect n in comparison with m. Using these we 

 obtain 



D0-D'6= *>-»') |jvpg -g+gfcj:. . (9') 



Equations (8), (9'), and (10) contain the solution of the problem. 

 Case 1. Let n = n ! ; (9') and (10) give 



X(« 2 + 6 2 )=>|r^+6 2 )J 

 Now D': D = //, 2 ; so that, from (8), 



or since 



2 



^Sigr-ti^tp ( 13 ) 



L« « ft +1 J 



={^+^^(^-1)}*,, • • • • < 13 > 



if we put 



4="!=M (14) 



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