a Theory of the Collapse of Boiler-flues. 229 



we get 



<r 3 =A/(ra + n), -\ 



(^\ = A.>b(- 2 U ' ' ' (11) 



\dr J m + n \dr /' J 



From (7) we get 



= K(^.) +<rs } 



_ 1 $ d™ j, A _ w _ 1 dhv\ .-„* 



"" a L<^> m + w a a dcf> 2 J 



If G be the flexural couple about a generator, measured in 

 the direction in which the curvature diminishes, 



G=- f Q'h'dh'. 



But 

 Q' = (m + n)o-' 2 + (?n — w) cr' 3 



_(»-«),, + {(*+») (J) + («-„) (*£•)} A' 



L (m + n)a\d<p m + n a adept J J 



accordingly, 



3 x 3(m + n) a \d<f) m + n a a a</>v v ' 



Now vrh? may be neglected when the surfaces of the shell 

 are free from tangential stresses ; and A Y h 3 and Ah 9 may be 

 neglected when the surfaces are free from normal stresses or 

 pressures. Under these circumstances we obtain 



~ Smnh 3 (d 2 w \ ... .. 



6(m + n)a i \dcj) 2 y K ' 



which is the well known result in this case, and shows that 

 the flexural couple is proportional to the change of curvature. 

 But when the surfaces of the shell are subjected to external 

 and internal pressures U ly n 2 , the terms involving A, A x 

 cannot be assumed to be negligible when multiplied by h*. 

 The quantity A is the value of R at the middle surface of 

 the shell ; and since R varies from — Ti 2 to — TTj, as we pass 

 from the interior to the exterior of the shell, it is evident 

 that A must have some value intermediate between these 

 two quantities. It therefore follows that under these cir- 

 cumstances the expression (14) is erroneous ; the correct 



