696 Mr. R. V. Southwell on the 



not concerned with terms of higher order than /* 3 , and a is 

 given, with sufficient accuracy, by (26). 



When X is zero, the wave-length of the distortion, in the 

 axial direction, is infinite. In this case ft must be given by 

 the formula (15) for a flue of infinite length, so that in the 

 notation of equation (26) it is clear that we may write 



_ 2 m 2 (P-l) . 



£?=- — 2 — rlL ~ — j— ^ + (terms involving powers of X). 



Now we have already seen that X must be very small in 

 any practical case, and it is therefore clear that the unknown 

 terms do not form any appreciable part of ft ; hence with 

 sufficient approximation we may calculate the critical value 

 of the pressure difference, in the type of distortion specified 

 by (23), from the following formula : 



The question now arises : — in the problem of the boiler- 

 flue, where distortion is wholly or partially prevented at 

 certain sections by the agency of " collapse rings," what is 

 the connexion between A, and the distance of these rings 

 (or the effective length of the flue) ? A purely ideal type 

 of constraint may be imagined, which tends merely to main- 

 tain the circularity of the tube at certain sections, without 

 restricting the slope of the tube-wall * ; this would permit 

 the occurrence of the distortion specified in (23), and the 



distance between constraints would be equal to — . But in 



A, 



practice the end constraints will also tend to " clamp w the 

 ends of the flue, and this effect will strengthen the tube, by 

 an amount which it is not easy to determine exactly. 

 In any case we may say that 



, 1 



/GC x' 

 <* a - («y>. 



and we may illustrate the way in which the end effects die 



out by plotting (IIx— II 2 ) against the quantity — -, or -. To 



clX q 



do this we must take some definite value of the ratio -, and 



a 



* A thin disk, inserted in the tube but not fastened to it, would havo 

 approximately this effect. 



