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

 a supposition will be, for each of the prisms, represented by the expression 



and therefore, for the whole element, by 



Ae" X number of prisms. 

 But the number of these prisms, being directly as the volume of the element and 

 inversely as the volume of each prism, will be represented by the expression 



Hence the error in the foregoing division wiU be for each element 



Now since the bounding planes of these prisms, taken two and two, are sym- 

 metrically situated with regard to the molecules which they contain, if the extre- 

 mities of the prisms were symmetrically situated with regard to the extreme 

 molecules, the number of such molecules contained in these two prisms would 

 evidently be as their lengths. But these extremities can always be made sym- 

 metrical by adding to one of the prisms a portion whose length is of the same 

 order as the molecular distance. Hence the error involved in the assumption, 

 that the number of molecules in each prism is proportional to its length, is re- 

 presented by an expression similar to 



The total error for each element is, therefore, expressed by a quantity similar 

 to Ce^'"'^'. Let I, I', I", &c., be the lengths of the several prisms into which the 

 element v is divided, and let X, A', X", &c., denote the corresponding quantities 

 for v'. Let also w be the common transverse section. Then it follows from 

 the assumptions which we have made, that 



■S:,m=^E(l + r+l"+&c.), 



^,m=E'(X + \'+X"+&c.). 

 We have also 



and therefore. 



