504 



MH. A. E. H. LOVE OX THE SMALL FREE VIBRATIONS 



3 3 3 "I 



^ =- (TO n 8 + 2ne) + h z ^ (nc) + h s ^ (nb) = 0, 



h^ . (?ic) + h 2 _- (m n 8 + 2n/) + h s ^- (no) =0, \ 

 h i *- ( nb ) + h * ^ ( na ) + ^3 5; (m-n 8 + 2n. 9 ) = 0, 



where m = k -\- 

 Hence, 



and 

 Thus, 



k being the modulus of compression, and n that of rigidity. 



= ' = ' 



\ 



- *) 



n ic a X, r 3 

 T+ n V ^ 



If there be no surface-tractions on the surfaces initially parallel to the middle- 

 surface, viz. , r = A 3 A, then A = 0, and B = 0, and also at the surfaces 



(m n) 8 + 2ng = 0, 



so that 



Thus, 



u = 





Ag m + n' 



K 



o- 2 





J 



Hence, 



* Expressions equivalent to these have been given by ARON, but his work contains an error. His 

 equations (7, a), (7, b), p. 145, are strictly analogous to equations (6) and (7) above, but the terms in 



p ~-(t~) are all omitted. The test * ^- = ~ ~ is not applied ; if it had been, there would 

 0* \."i/ dp Og v' Sq Op v' 



have resulted equations which in my notatiou are T\ = 0, T' S = 0, but the values of t\, T' S are calculated 

 subsequently by the method of Art. 7, and are the same as those given in equations (8). 



