188 Mr. Nikhilranjan Sen on a Type of 



Similarly, the second relation is replaced by 



A f ^ sin ha j d 2 sin ha ~| 

 \_dlia ' ha d(Jia) 2 ' ha J 



R |"(i cos 7i« , d 2 cos 7ia , " i 

 \_dha ' ha d(Jia) 2 'ha J 



Eliminating A, B, C from these equations, we have the 

 frequency equation in the form of the determinant (denoted 

 b} r the leading diagonal terms) 



(ai6 s -c 8 ) = 0, 



where 



r , i / d 2 2\ <#\r/sin#\ /cos^\~I 



[«„ y = (,,~ 2 + ^-^ . r J (_(---) , (— )j ; 



r , -, r d 3 2A, d /I rfVl r/sin#\ /cos#\~] 



[a * ^ = L^ 3 + xr^ ■ a ( s ^JJ Ht) • {—)] • 



r , -, /d d 2 \ r/sin a< \ /cos#\~| 



[o to 6J = ( 5 +« 3? )[(— ), (— )J; 



c 1 = 0, c,= (l_e)n', C3 =(l-26) 7 2 + i(l-e) 7 ^ 

 x — ha, n' = c 2 p/(\+2/ju)e. 



The quantities a l5 6 X are to be obtained when the operator on 

 the right-hand side of the corresponding equation operates 

 on (sin#)/# and (cosa?)/# respectively, and similarly for the 

 other elements of th© determinant written in this manner. 

 Neglecting e in comparison with unity, the frequency 

 equation may be written 



(y 2 -\-iyx)(a 1 b 2 — a 2 b 1 )—?i'(a l b 3 — aJ) 1 ) = 0. 

 Multiplying out, we have 



(a ± b 2 — a 2 b l ) = { (3 v — v 2 ) — x 2 } /^ 3 , 

 (aj> 3 — a i b 1 ) = {l — e(2— v)}/x, 

 where 2\/(X + 2{i) = 2 — v = 2<r/(l — <r), 



a being Poisson's ratio. 



Neglecting (2 — v)e in comparison with unity, since (2 — v) 

 is a positive quantity less than 2 for all material bodies, the 

 frequency equation becomes the cubic 



ivx* + (ry 2 + n')x 2 -i(3v-v 2 )yx-(3v-v 2 )y 2 = 0. 



