j L. B. Tuckerman 



Then: ^ +i=x _^_ ?flP (^ 0ft+1 — N Elc+ j 



where .Af. + 1 is the differential order of the plate and is positive if the 

 ordinary wave is the slower. 



By forming the expression O k+l E k+l cos<£ i + 1 ; expanding in terms 

 of x an d *p; substituting the values found above for O k+i cosi/', 

 O k + 1 s'mvj/, E k + 1 cosx and E k + l sinx', and simplifying the resulting 

 expression, the following equation is obtained: 



O k+1 £ k , lC os<t> k+ =[y 2 (£\-0\)sm2(/cJci-i) 



+ O k E k cos 4> k cos 2(/e,fc-\-i)~\ cos27r7V A . +1 -(- 0,.E k s'm4>,.sm2irN k+1 . 



Similarly : 



O k+1 £ k+1 sm<t> k+ =-[y 2 (E\-0\)sm2(k,/c+i) 



+ O k E t cos<f> k cos2(k,k+i)] sm27rN k+l +O k E k smcf> k cos27rN k+1 . 



To simplify the above results the following notation is intro- 

 duced : 



2 =I I+J=2P OEcos<f>=K 



E 2 =J I~J=2Q OEsin^S 



Introducing these symbols the equations become : 



I k+ =I k cos\k,k+i)+J k sin>{k,k+i)+K k sm2{k,k+i) 



/, + 1 =/,sin 2 (/t, k+ i ) +/,cos 2 (k, k+ 1 ) — A",sin 2 {k,k+ 1 ) 



K* + i=l%(J— / k )sm2(6 > k+i)+K k cos2(k,k+i)]cos2irN k+l 



+S k sm2irJV k+1 , 



S k+ . 2 =-\_y2Q k -/ k )sm2(k,/c-\-i)+K k cos2(/e,k+i)^sm27rN k+1 



+S k COS2irlV k+1 



and finally the inductive theorem in its first form : 



Q* +1 =+Q ft COS2(v&,/fe+l) 



4-K k sin2(k,k+i) 



(2) 

 K k + l = — Q k sm2{/c,k J r i ) cos27ryV A . + 1 



+K k cos 2 (/cjc^r 1 ) cos 2 ttA\ +1 



+ 5 A .sill27rA^ ; + 1 

 l6o 



