226 Mr. A. B. Basset on the Difficulties of Constructing 

 whence the last two of (2) become 



dx 1 dx 





«'»(* + S) 



dv . at » dv 

 dx* dx 



whence, eliminating N, we have 



V co/dx 4 co dx 2 dt 2 dx 2 dt 2 



which is a well-known result. 



Let I be the length of the rod ; then we may take 



v oc e ipt+imxl1 , 



where m is an integer. Substituting in (3), we obtain 



/_, <™ 2 m 2 7r 2 \ . mW< I , TA/AwV , T x ) ... 



<i+-7-V=T-{^ + »hp- + «/- • (4) 



When 1\ is positive, so that the force is a tension, p is 

 always real; but when T x is negative and equal to — P, so 

 that the force is a pressure or thrust, p will be imaginary 

 unless 



P< 



K 2 coq 



K 2 + l 2 /mW 



The least value of the expression on the right-hand side 

 occurs when w=l ; the condition of stability is therefore 



If the length of the rod is large in comparison with the 

 radius of its cross section, the term tt 2 k 2 /1 2 in the denomi- 

 nator may be neglected, and w r e obtain Euler's law of 

 thrust. 



7. We must now return to the subject of boiler-flues. 



This problem has recently been discussed by Mr. Bryan by 

 means of the energy method*; but his work, as distinguished 

 from his result, is vitiated by the assumption that the potential 

 energy per unit of length of the cross section due to bending 

 is equal to 



which, as w r e have already pointed out, is only true w r hen the 

 surfaces of the flue are free from external pressure. 



* Proc. Camb. Phil. Soc. vol. vi. p. 287. 



