224 Mr. A. B. Basset on the Difficulties of Constructing 



the condition of stability requires, that the periods should 

 be real quantities. In the second place, let the potential 

 energy in the deformed state be found ; then the condition of 

 stability requires, that the potential energy of the flue when in 

 equilibrium should be a minimum. Either of these methods 

 will determine the mathematical form of the function F. 



In order to apply the first method, it might be supposed 

 that we might start with the general equations of an elastic 

 solid, and calculate the values iv , v of the radial and tan- 

 gential displacements when the flue is in equilibrium. Having 

 done this, let iv + ii\, v + vi be the complete values of the 

 displacements when the flue is performing small oscillations. 

 The values of all these quantities can be obtained from the 

 equations of motion ; and by means of the boundary con- 

 ditions, some of the arbitrary constants which occur in the 

 solution can be determined, and the rest eliminated ; and the 

 resulting equation will give the period. 



But in attempting to apply this method, it will be found 

 that the pressures Tl 1: TT 2 disappear; consequently the periods 

 of vibration are the same as the free periods, when the flue 

 is not subjected to any surface-pressure. This result is ob- 

 viously wrong. The reason is, that the general equations 

 of an elastic solid in their ordinary form are linear. In 

 order to solve the problem by the first method, it would be 

 necessary to take account of certain quadratic terms, in wdiich 

 quantities upon which the motion depends enter into com- 

 bination with the surface-pressures. 



The problem may be illustrated by considering the pro- 

 pagation of w r aves in a liquid. When the liquid is initially 

 at rest, all quadratic terms are to be neglected ; but when 

 the liquid possesses an independent motion, all quadratic 

 terms which depend upon this independent motion must be 

 retained. 



The application of the second method involves a knowledge 

 of the expression for the potential energy due to deformation. 

 The form of this function is known when the surfaces of the 

 flue are free from external pressures*; but this expression, 

 as we have already pointed out, is inapplicable when there 

 are external pressures. 



6. In order to understand more clearly the necessity of 

 retaining the above-mentioned quadratic terms when applying 

 the first method, let us consider the stability of a rod of 

 length I which is subjected to a tension T v Then employing 

 the notation and method explained in my 6 Elementary 



* Phil. Trans. 1890, p. 443, equation (24). 



