212 The Rev. J. H. Jellett on the Eqii'dibrium and Motion of an Elastic Solid. 



Let us now consider the order of the error introduced by each of these sup- 

 positions. 



(1.) The supposition that u remains the same for all molecules situated within 

 the element «, will introduce an error whose order is the same mth that of the 

 actual variation of u within that space. We assume here, that the function u 

 varies continuously within the space (o ; in other words, that if u>' be divided 

 into any number of equal parts, the variations which u receives in each of these 

 parts are quantities of the same order of magnitude. Hence it is easily seen, 

 that the variation of u within the space k"e'" will be represented by an ex- 

 pression of the form 



F'V"-'. 



For if w be divided into a number of parts, each equal to k"e'", the variations 

 in these segments may be represented by 



the exponent m being, in conformity with the foregoing assumption, the same 

 for all. Hence we shall have for the complete variation of u, 



(a, + a2 + &c.) e". 

 Now since fli , 02 . &c., are finite quantities, 



ai + a2+ &c., 

 will be a quantity of the same order as their number. Denoting this number 



by n, we shall have 



' 7.' 

 _ " _ ^ j--i- 



Hence it is evident, that the complete variation of u is of the order 



m + i' — i". 



Since, therefore, tliis variation is by hypothesis finite, we must have 



m + i'-i"=0, 

 or 



m = i"-i'. 



Hence the ejqjression for the partial variation of u is 



k'"/'-'. 



