520 MB. A. E. H. LOVE ON THE SMALL FREE VIBRATIONS 



3*10 1 3V /, 3*10 



- ' + 





, _ l - r ~ 



3/3 [A, \ m l /\ r A,3/3\ m 



3 fAi /.., m 7i \"1 1 3A, / , m n 

 - -L- 1 ( 2X, -- -K, H r ~ ! (2X. -- - 

 alAj\ m f /J A s 3a V m 



3 / 1 \ / m - 



XV - /xU + I nh* -- p* 2* 3 - - X, - X2X 1 - K > + 2 V *i = 0- (35) 



r v/ 



TO 



The first terms in each of these equations are the same as those in CLEBSCH'S 

 equations, pp. 306, 307. 



The couple [XU + /tV] is that called by DE ST. VENANT the moment of torsion ; 

 the couple XV /j,U is that called by him the moment of flexure, and their axes are 

 the normal and tangent to the edge respectively. The former of these may be con- 

 sidered as arising from a distribution of force in lines normal to the middle-surface 

 and in the edge ; the difference of the forces in consecutive elements gives rise to a 

 resultant force normal to the middle-surface which coalesces with C. This is the 

 explanation of the union of two of the boundary-conditions given by POISSON in one. 



We are going to apply the equations just developed to determine the small free 

 vibrations of the shell. The terms depending on the rotatory inertia will be 

 neglected. 



6. Possibility of Certain Modes of Vibration. 



1 2. Now let us suppose, if possible, that the shell vibrates in such a manner that 

 no line on the middle-surface is altered in length. This requires that <r lt cr 2 , CT be all 

 zero. Thus, from equations (13) we derive 



, 3w . , , 3 / 



