Mi; R. V. SOUTHWELL ON THE GENERAL THEORY OF ELASTIC STABILITY. 201 

 Thus we obtain 



aS= P 



1- 



2m- 1 <i 



/ m G Vi i m-1 } 



-\rn-2 4C/\ m+l l< ' 

 2 m-\ G 



-2)'2C ., . 

 at cotli at 



) = It 



( m G \ / , m-} G \ 

 ZT>> if V I , i ' T7i / 



at tanh at 



aS = P 



WI- 



TH 2 \ wt 



40 



at tanh at 



Vm-2 4C' 



aQ = R 



n 



4(, 1 



wi G 



- I at coth at 



^w 2 



(28) 



There are two solutions of the equations (28). Either 

 2m -1 G 



1- 



(m+l)(m-2) 2C , aO .,T-I/ 4(- \ ,, 



at tanh at = - p = 2 - T I at coth at, 



G\7, . m-1 ( R m-2\ wt G I 



and 

 or 



1- 



ro+1 4(J/ 



2m -1 G 



-2 4C/ 



P = S = 0, 



(29) 



-2 2C 



aS m-\ 



Ijn _G 



and 



F *\-S A Al A C*kJ ^ III I 



-PTT at coth at = -=^- = 2 - 

 -1 G\ P wi-2 



, _ 

 4C 



. m . ^ 

 f mTT ' '" 



m ( 



at tanh at, 



(30) 



The criterion for neutral stability is in the first case 



r-1 G- 



1 + 



,/ .1 k A 



at (coth at -tanh at) = 



and in the second case 



1 + 



1 + 





4 . v 

 at (tanh at-coth at) = 



1 + 



2i(m+l) C 



2m"- 1 G_ 

 2>n(wi.+ l)' C 



1 < 



2wi(m + l) C 



Vi >L. CCX1IJ. A. 



2 D 



