RUPTURE OF THE VESSELS WHICH CONTAIN THEM. 117 



6. Should the interior force be a vibration of the kind already explained, and 

 should the exterior action be extremely feeble, and act on a very great mass, 

 this extremely feeble action may crush the vase inward, with a power that 

 shall exceed, in any degree, the enormous action of the interior and explosive 

 vibrations. The comparison of the interior and exterior actions is best effected, 

 in this case, by finding the modulus of elasticity of a material spring that shall 

 coincide most nearly in effect with the interior tremor. For, putting e and e' 

 respectively for the modulus of the spring and of the support, and « and a for 

 the deflections resulting from the tremor acting alone, and the reaction as it 



does act, we have — = ^ -^ , or, in other words, the deflection produced by the 



reaction, is to the deflection that would be produced by the interior tremor 

 alone, in the inverse proportion of the square roots of the moduli of tremor and 

 support. 



7. Combining what is here said with the known laws of fluids moving in 

 pipes, and whereby they necessarily produce hydraulic shocks, it follows that 

 any vessel connected with such a train of pipes, and plunged at some little 

 depth in a considerable mass of water, or other heavy fluid, will occasionally 

 be subject to a crushing and exterior force vastly greater than the interior strain 

 due to the constant head of fluid. 



To investigate these results, let us commence with the very simple case of a 

 mass m, urged by two moving forces^ and — f , which tend to deflect it from 



a point of initial repose. The vis viva in this case will be J ^ {fds — f'ds} 

 where s is the distance through which the mass has moved ; and as this vis 

 viva must be zero when the mass has attained its limit of deflection, we have 

 for that case 



Now it is clear that if the forces f and — f acted only at the position of 

 load, we should have simply 



/ = /'; («) 



and consequently if we express by g and s' the deflections occasioned by an in- 

 stantaneous and an accumulated load, resulting from the action oif — f when 

 commencing at the initial position, and the position of load, we must determine 

 the first from equation (1), and the second from equation (2). 



VII. — 2 E 



