a Theory of the Collapse of Boiler-flues. 233 



Accordingly from (18), (19), (20), and the last of (16) and 

 (17) we get 

 {Ia- 1 (D 2 + l) + (n 1 -n 2 )}(D 2 + l)D 2 ^+ / .a(D 2 -l)^ = 0. 



To solve this assume 



w a gljrf + M^ 



where s = 2 . 3 . 4 . . . , then 



{ia-i^-ij-lcni-u^^-iy^^+i)^ ; (2i) 



and therefore p will be real provided 



Ili-II^I^-lJ/a. 



The least value of the right-hand side occurs when 5 = 2, 

 in which case 



n 1 -n 2 <3i/a 



. 8A 3 / nv, 



a 3 \?72 + ?2 



+ 



«) (22) 



This is the condition for the stability of the flue. 



In the particular case considered by Mr. Bryan, U 2 = 0, 

 and since it is practically certain that a is a linear function 

 of IIx, we may write a = HI 1 , where k is some constant which 

 is independent of h. Whence (22) becomes 



TT (l- S J^\ 8A 3 mn 

 \ ^ ) < a\m + n)' 



Since powers of h higher than the cube are to be neglected, 

 the condition of stability becomes 



TT 8h B mn 



a d (m-\-n)' 



which is Mr. Bryan's result. 



Although I am disposed to think that this result is 

 rigorously true as a first, and probably for practical purposes 

 as a sufficient approximation, yet a rigorous mathematical 

 theory requires a knowledge of the correct expression for the 

 energy. An inductive process, by means of which a function 

 is shown to satisfy all the requisite conditions, would prob- 

 ably be the simplest method of discovering its value. 



August 9, 1892. 



