MR R. V. SOUTHWELL OX THE CENEKAL THEOUY OF ELASTIC STABILITY. 233 



Eliminating T, and T, from equations (91), and substituting from (92) and (93), we 

 have 



u' ^ 3V ( 1 3t/ t 1 



a* 32* a 2 30 ma 3z 



t*[l 3V 1 8V 2 8V 3V 1 3V tii-l 1_3V_~| 

 3 La 4 30" a 4 S0 4 a* 30*32* 82* m* 32* " m * 3032*.] = 



I P^ 1 aV m-1 8V . /m+1 . \ 1 3V 

 a 30 a 1 36 s 2m 3z \ 2m ' /a 303z 



_^n<V ,13V 1 3V m- 1 1 3 VI 

 ~ 3 L<? 30 a 4 dP a* 30 82* m a* 82* J * 



J_ <to/ m + 1 1 8V m-1 1 8V 3V = 



o. o 2tf* a.** " ' 



82 " 2m a 3032 2m a* 30* " 82* 



(95) 



where 



(96) 



Assuming a solution of the type (56), we find, as the criterion for neutral stability, 



m a 

 m 1 i i w 1 , 



m ' a" 



m+1 ;, 

 2m 



m 



m + 1 

 2m 



m 1 



= 0. 



(97) 



This equation, in its expanded form, is 



t 4 + 3/ty + 2 



^=1 g 4 ]} = 0. 

 nr 



(98) 



Taking first the terms which are independent of t 1 , and neglecting the square of 

 (which must be small), we find, as the first term in our solution, 



m*-l 



m 



(99) 



VOL. CCXIII. A. 



2 H 



