210 The Eev. J. H. Jellett on the Equilibrium and Motion of an Elastic Solid. 



results which it furnishes are mathematically accurate. When the mass ceases 

 to be continuous, these results become approximate, and would of course be 

 valueless, unless we had some means of testing the degree of approximation at- 

 tained. This we shall now proceed to consider. 



Let rii be the mass of any one of the separate molecules of which the body 

 is composed, and let x, y, z be its co-ordinates. Let mf{x, y, z) be the mathe- 

 matical expression of some quality or power belonging to this molecule, of 

 such a nature, that the corresponding quality of the entire body is mathemati- 

 cally expressed by the sum of the expressions which refer to the several mole- 

 cules. Let also m', x', y\ z', m", x", y", z", &c., be the masses and co-ordinates 

 of the other molecules. Then, if we assume 



"=/(^)y. ")' u'=f{x',y',z'),&c., 



the accurate expression sought for will be 



mu + m'u' + m"u" + &c. = Smw. 



Now let dv be the element of the volume geometrically considered, and « the 

 mean density of the matter which occupies it, so that its weight may be repre- 

 sented by 



gedv. 



Then the approximate equation furnished by the integral calculus will be 



'Emu — J uedv. 



In order to estimate the amount of the error which is involved in the use of 

 the integral sign instead of the symbol of finite summation, we shall consider 

 successively the several suppositions which are made in the interchange of 

 these symbols, and the amount of the error introduced by each. 



The object of this investigation being to determine, not the actual magnitude 

 of the error, but merely its order, it is in the first place necessary to establish a 

 notation to represent the respective orders of the several small quantities with 

 which we are concerned. 



Let e be an indefinitely small quantity which we take as the standard. Let 



the distance w, between two consecutive molecules, be of the order i, or in other 



words let 



10 = ke'; 



