a Theory of the Collapse of Boiler-flues. 227 



Mr. Bryan has also attempted to introduce the external 

 pressures into his expression for the total potential energy due 

 to strain, by taking into account certain terms depending 

 upon the extension of the middle surface ; but this does not 

 help the matter, for if the pressures n i5 n 2 were suitably 

 chosen, it would be possible to prevent the middle surface 

 from experiencing any extension, but the expression for the 

 potential energy would still contain terms depending on IL 

 and TT 2 . 



8. The necessity for introducing the external pressures may 

 also be seen as follows. 



Let the flue be regarded as infinitely long, so that the 

 problem is one of two dimensions. Let a be its density, 2A 

 its thickness. Then the equations of motion are * 



dT N 



ITs + 7 =/u? ' 



§-M'-j)- n 'K)-« > ■ <6 » 



where p is the radius of tbe deformed middle surface, and 

 /jl=2Ii<t. 



Now if we knew the correct expression for the total energy 

 of the system and were to apply the Principle of Virtual 

 Work, we should obtain (i) two equations connecting the dis- 

 placements v and w ; (ii) the values of T, N, and G in terms 

 of v and w. And if we were to substitute these values of T, 

 N, and G in equations (6), the last one would reduce to an 

 identity, whilst the first two would reproduce the equations 

 of motion in terms of v and w, which we had already obtained. 

 Since the second of (6) contains the pressures n 1? IT 2 , it follows 

 that the expression for the energy must also contain these 

 quantities. 



From (6) it appears that the problem of the collapse of a 

 boiler-flue can be solved, provided we can obtain the value of 

 the flexural couple G, and provided also we assume that the 

 extensibility of the middle surface may be neglected. 



9. The expression for this couple in the case of a strained 

 cylindrical shell is given in my paper in the ' Philosophical 



* If the extensibility of the middle surface is taken into account, the 

 right-hand sides will contain certain additional terms — see Phil. Trans. 

 1890, p. 439, equations (11) ; but the existence of such terms does not affect 

 my argument. 



