The Rev. J. H. Jellett on the Equilibrium and Motion of an Elastic Solid. 213 



Let Ml be the least value of u within the element v, and u^ + k"'e'"''' the 

 greatest, and let it be supposed, as the most unfavourable case, that u has 

 throughout the value 



Substituting this expression in l^mu we have 



2,wm = MiSiW + k"'e'"~'' SjWi. 

 Hence the error in the equation 



2i7?lW = MiS,?/* 



is at most 



and, therefore, the error in 



'^mu — ?<,SiOT + Ui'Zjn + &c., 

 is, at most, a quantity of the form 



This equation will, therefore, be free from sensible error if 



(2.) In estimating the error produced by the second supposition, we shall 

 assume that the densities and magnitudes of the molecules vary vnth a finite 

 degree of rapidity ; and that, therefore, at any one point in the body, the sum 

 of the masses of the molecules contained in an element is proportional to their 

 number. Hence the equation 



is equivalent to an assumption, that the number of molecules contained in tlie 

 element v is proportional to its volume. 



To estimate the error involved in this assumption, let us compare, for the 

 sake of greater generaUty, two elements whose bounding surfaces are wholly 

 different in form. Suppose these elements to be similarly divided into rec- 

 tangular prisms with the same transverse section, whose linear dimension is 

 of the same order with the molecular distance. The error involved in such 



2 f2 



