MEMBKANES AND PLATES 



143 



Thus by superposition of the modes for which s = 2, s' = 1 and 

 s = 1, s' = 2, respectively, we get 



oe sin 



.Try . irx . 



sm - -- + X sin sm 



a a a a 



. TTX . iry ( TTX Try\ , . 



sm sm - - cos h Xcos -1 , (6) 



a a \ a a J 



where X may have any value. For example, in the cases X= 1 

 the diagonals a; + y = a, x y = 0, respectively, are nodal lines. 

 The figure shews the cases X = 0, X = , X = 1, which 

 give a sufficient indication of the various forms that may 

 arise. 



A"i 



Fig. 49. 



Again, by superposition of the cases s = 3, s f = 1 and s 

 s' = 3, we get 



. OTTX . fry . TTX . tTry 



sm sm - -f X sm sm 



a a a a 



a a ( a V a )) ' 



The cases X = 0, X = J, X = + 1 are shewn in Fig. 50 ; 

 intermediate forms are readily supplied in imagination. 



A still greater variety is introduced by the fact that a 

 number which is the sum of two squares can sometimes be 

 so resolved in more than one way. For example, the modes 

 for which 



s = 4, 7, 1, 8,| 



s = 7, 4, 8, 1,} 

 respectively, have all the same frequency. 



