Mil. R. V. SOUTHWELL ON THE GENERAL THEORY OF ELASTIC STABILITY. 217 

 m A B . A~| . i m A B 



- f kqrr A.;,, -f ^kqtr^ - $kf/ - + <T 



[_m 2 4 4 J 



+ <T , 



(78) 



We may now eliminate the coefficients from equations (64-78), and obtain a 

 detciinin.uital equation, of fifteen rows, which gives a relation between A, B and the 

 dimensions of the tube. This relation is the condition for neutral equilibrium of the 

 initial stress-system, and is clearly correct to terms in T* ; but by further consideration 

 of the terms involved we may show that the labour which would be required for its 

 complete evaluation is. unnecessary, and as the fifteen-row determinant may be 

 written down directly from the above equations it will not be given here. 



Solution for Boiler Flue tvithout End Thrust. 



We shall begin by deriving a sufficiently approximate expression for the difference 

 of pressure required to produce collapse of a tube, when there is no resultant end 

 thrust or tension ; and in the first case we shall deal with a form of collapse possible 

 only in the case of tubes of infinite length. That is to say, we make B and q equal 

 to zero in the fifteen-row determinant, which may then be reduced to one of ten 

 rows. 



In the latter determinant we may treat A as a quantity of order T* ; for if A be 

 put equal to zero, and the determinant be expanded, the terms which are independent 

 of T vanish identically. Hence A may be written for o-A, and AT* may be 

 neglected. 



The ten-row determinant, simplified by these and other obvious modifications, is 

 given on pp. 218 and 219. Expanding it from the top row, with the neglect of terms 

 of order higher than r 2 , we obtain 



m 2 \m ' 

 whence 



VOL. CCXIII. A. 2 F 



