AND DEFORMATION OF A THIN ELASTIC SHELL. 521 



These are three partial differential equations to determine the forms of u, v, w; and, 

 if either u and to or v and w be eliminated, they will in general lead to an equation 

 of the third order to determine v or u. When one of these is determined, the rest 

 are known. But at the edge we have to satisfy four boundary-conditions, and this 

 will not be generally possible with solutions of a system of equations such as the 

 above. 



13. Since <r lt o- 2 , CT may not in general be regarded as of a higher order of small 

 quantities than * 2 , X^ K lt it follows that the term in W z in the potential energy 

 which contains h as a factor is very great compared with the term in W t which 

 contains h 3 , and we may form approximate equations of vibration and boundary- 

 conditions by omitting the latter term. 



The equations of motion thus formed are 



* , L T a f 1 / 2m 

 = vi + M* ~ 2 a~ 1 T - 



& PL * IA \ m + n 



8/l\/2wi, . m-n 



2m . m n \ , 9 / w \" 



l > 



n \ 1 2m nin 



And the boundary-conditions are 



. / 2m .771 TO \ 



2X - - <r, + - - <r 2 -f /ACT = 0, 

 Vm + n r 7/n-n V 



/2m m n \, x 



2/x - <T 8 -f - o-j + XCT = 0. 



\m + n m + n 7 



(1.) Let us examine the possibility* of purely normal vibrations. 

 Since u = 0, v = 0, the equations of motion become simply 



' '' , r. n 2771 /I 1 2<7 \ /qo\ 



37. + 2- -{ t + ri-K: )w = o, ..... (38) 



d< 8 /) TO + n \p l * p? ptfiJ 



-where <r = (m n) /2m is the ratio of linear lateral contraction to linear longitudinal 

 extension of the material of the shell. 



* MATHICI: convince!* Limself of the impossibility by general reasoning. 

 MDOCCLXXXVIII. A. 3 X 



