304 Mr. A. Gr. M. Michell on the Elastic Stability of 



V. If instead of a single transverse load a uniformly distri- 

 buted load be applied, the equations are still simple. Consider 

 the case of the cantilever as in the first case treated. With 

 all the other symbols as before, let p be the load per unit 

 length. Then 



whence 



also 



B 2 =+| X 2 0; 



dx ~P xU+ 2 a: dx' • 



dx ~ 2 dx 2 ' 



Thus 



but as before 



Therefore 



or 



d 2 6_dO__ B d?y 



dx 2 dx 



dx 2 



dx 2 ~ A fit Dl fiS. 



A- A 



0. 



L P, 2 A §'. 



Ll AA 



d?0 



dx 2 



2 0=0, 



d>0 ffr-fa 

 dx^ + 4 AAy 



If for p be substituted P/7, P being the total load on the 



cantilever, T>2 ,^2 a \ * s a 1 uan tity of the same kind as J, 

 r (p, + p 2 ) 



and may be denoted by H. 



The differential equation is then 



d 2 



*0 



dx 2 + urn ~ ' 



and the solution in series 



->.6 



^~ A { 1 (4m 2 ) 6. 5 



+ 



(4H/ 2 ) 2 . 12 . 11 . 6 . 5 



+ 



(4HZ 2 ) w 6n(6rc-l) {6n-6) (6n-7) ...6.5 



■■} 



