522 MR. A. E. H. LOVE ON THE SMALL FREE VIBRATIONS 



In order that all parts of the system may be in the same phase, it is necessary th.it 

 1/pj 2 + l/p^ + 2<r/pip. 2 = const, all over the surface. 



Again, in the u, v equations we must pick out the terms containing w, and, observing 

 that w is independent of a, ft, we may write them 



3* IA \Pi Pa /I d* \hj \p 3 pJ 



Thus, 

 and, 



But, by equations (17), 



' ' - ' 6 ~ L )IS 

 Substituting, we get 



= 



i g Y :^>'v- :- (39) 



A| as \p, "*~ PO J = ' J 



So that 1/pj + l/p 2 =r const, all over the surface. 



The two conditions of possibility of normal vibrations show that the middle-surface 

 must have both its principal radii of curvature constant at every point. These 

 conditions are satisfied by the sphere, the circular cylinder, and the plane. 



Again, if the surface be bounded by an edge, we have, since vr=-Q, \(l/p l + cr/p^= 0, 

 /* (Vpa + "//i) = J these can coexist for all values of X, fi if tr 3 1=0, and 



I/Pi = I/ft. 



To make n positive, or the material resist distortion, we must have ^ <r positive, 

 so that a- cannot be = 1 ; the equation cr = 1 makes n = 3m = 3k + n, so that 

 k =. or the material of the shell would offer no resistance to compression ; thus, the 

 equations above written cannot coexist for all values of X, /A, and hence one of the two 

 X, /x, must be zero, and one of the two equations l/p 2 + <rfp\ = and l/p t + cr/p. z = 0, 

 must hold at the edge. These conditions cannot be satisfied on a sphere or cylinder. 



