300 Mr. A. G. M. Michell on the Elastic Stability of 

 or from the equations (2), 

 Pi 



N da* * & 



{--<;-*&}/ 



dz 



fa d 2 y _ fa -fa ff 



7 dO dy , N 



The first term on the right of the first of these equations 

 is finite, the others are infinitesimal and may be rejected. 

 The two terms on the right of the last equation are of course 

 of the same order and must both be retained. Comparing 

 the first two terms of the second with the corresponding 



terms of the third equation, it is seen that I -~ j J is of the 



di/ 

 same order as y//3 2 , so that -j- is either of the same or of 



higher order than 0, and therefore (y—yi) of the same or of 

 higher order than x0. Hence the last term of the second 

 equation can be rejected, and the equations become 



= m, 



fadh^ 



fad 2 y_ fa-fa 

 N da; 2 fa 



7 d0 __ x dy +( s 

 N dm" dx +{y yi) - 



x9, 



(3) 



These equations may be used with two distinct assumptions 

 as to the orders of the quantities. 



Firstly. If 7 is of the same order as fa, j- of the same 



order as 0, and ^ small and of the order of ^- . 



dv 

 Secondly. If 7 is small compared to fa, -j- is to be of 



higher order than 0, and fa fa may be of the same order, 



-^ of the order of -~ , and 3- and ~- of the order of 

 dx dx. fa p 2 



(-1 



\dm ) 



From the last of equations (3) 



<P6 N d*y 



dx* ~ 7 X dx» 



