190 Mr. Nikhilranjan Sen on a Type of 



roots real, or have a pair of imaginary roots according as the 

 equation ~D(ai)=0 has all its roots real or has a pair of 

 imaginary roots. It is to be noticed that in the former case 

 there is no vibration, the initial state of strain in the shell 

 disappearing gradually with time without causing oscil- 

 lations, giving rise, at the same time, to the corresponding- 

 types of disturbance in the surrounding atmosphere. 



The limits within which such a state of motion would 

 exist can be calculated thus. Putting in the values of 

 « , a u a 2 , and a %} the discriminant can be written as 



D(n') =4m; i /3 /27 7 2 - {£ f m?/y 2 - %m)n n 



-(Mm 2 -|m 7 2 V+(/ T m 3 +^iy+2 4 T W7 4 ), 



where n' =pc 2 l(\ + 2fi)€ is a positive quantity; for many 

 ordinary material bodies and air m is nearly 2, and 7 is a 

 fraction whose average value we may take to be a fraction 

 of the order of y 1 ^. Neglecting the smaller terms we have 



approximately. It has been shown that the period equation 

 has all its roots real if the equation D(?i') = 0, a cubic in n' 

 has only real roots, in which case the discriminant of the 

 equation D {n') = should be negative. Moreover, the values 

 of n' which make D(n') negative, and consequently make all 

 the roots of the period equation real, are those which lie 

 between the positive roots of D(n') = 0. It may be shown 

 that the discriminant of D(n') =0 is negative if 



J ff (47-m/ 7 2 )-500 7 2 /m<0, 



considering only the approximate form of the equation. As 

 this condition is realized for almost all material bodies and 

 air, there is for a shell of every such material two critical 

 thicknesses, within which we should expect the shell to get 

 rid of the initial strain without performing vibrations. The 

 critical values are given by the two positive roots of the 

 equation D(?i') = 0. This analytical result, however, it is 

 very difficult to interpret physically, but it must be observed 

 that this is a necessary consequence of the reduction of the 

 period equation to a cubic. 



6. 



The numerical calculations for a particular case are given 

 below. If the material of the shell be glass and the outside 

 atmosphere be air, we take 



p—0013, 8 = 2-5, X = 2-15xlO n , ^ = 2-40 x 10 11 , 



and c = '332 x 10 5 cm. per sec. 



