The Rev. J. H. Jellett on the Equilibrium and Motion of an Elastic Solid. 211 



where k is a finite magnitude. Let i' denote the degree of rapidity with which 



the function u varies, i. e. let it be supposed that u receives a finite increment 



in passing from one to another of two molecules, whose mutual distance <«' is 



given by the equation 



m' = k'/. 



Suppose now the entire geometrical space which is occupied by the sys- 

 tem of molecules, including also the small intervals or pores which separate 

 them, to be divided into an indefinite number of equal portions, v, the linear 

 dimension of each of which is a quantity of the order i". We shall then 



have 



V = k"V". 



Let 2i mu denote a finite summation extended to all the molecules contained 

 in the first of these elements, 22mM a similar summation for the molecules of 

 the second element, '2377111 for the third, &c. Then 



Swim = 'S.^rnu + 'Z^niu + '^77iu + &c. 



This equation is evidently exact. 



Now let the following suppositions be made : 



(1.) That u retains the same value for every molecule within the ele- 

 ment V. 



(2.) That the coefficient e, which represents the mean density, is indepen- 

 dent of the magnitude of the element. 



These suppositions will give the following equations: 



S, mu — Ml 2, 7)1 = M, £] V, , 

 'S.-^mu = u^'^.^m — U2e2V2, 

 &c. &c.; 



"Sfinu = UiSiVi + u-i^iV-i + &c. = Swey. 



Finally, instead of the symbol of finite summation 2, let us substitute the 

 symbol of integration |, and we shall have 



2??iM = \tcedv = \udfi.. 



VOL. XXII. 2 F 



and, therefore, 



