a Theory of the Collapse of Boiler-flues. 225 



Treatise on Hydrodynamics and Sound/ ch. viii. p. 147, we 

 have 



whence 



Accordingly 



A'B' T, 



AB qco 



fa + T 1 /o.)(l + /i/p') _ 

 i_ " 1 + h/p 



whence 



o V co/\p l pJ 



When no forces other than stresses appHed to the ends of 

 the rod act, the equations of motion — § 135, equations (7) — are 



dT 



N 



ds 



P'~ 



ds 



T 



«5, ^ (2) 



^ .XT 2 I 



as T J 



When the rod is naturally straight and is free from stress, 

 T, N, and Gr are zero in equilibrium ; consequently, when the 

 rod is performing small oscillations, these quantities will be 

 linear functions of the displacements and their differential 

 coefficients ; accordingly N/p\ T/p' are small quantities of the 

 second order, which are to be neglected. But when the rod 

 is subjected to a tension T 1? the value of T is equal to T x + T', 

 where T' depends upon the motion ; accordingly the quadratic 

 term T/p' cannot be neglected, but must be replaced by TV//. 



Writing —dx = ds, and recollecting that 



dcP V- dx } 



1 d?w 



P' 

 (1) becomes 



r „ / TAdV 



