GENERAL PRELIMINARY RESULTS. 25 



and consequently 



[j , v> du> t* 

 - IdydzV -j- = - d 



} dx ] 



' dx 



v - -j . 

 w dx 



Then, since the sum of the elements represented by da-'j toge- 

 ther with those represented by dcr", constitute the whole surface 

 of the body, we have by adding these two parts 



[j J fjr dU " T7' dU '\ fj dx 



\dydz V -! -- - F j = \dcr-j- 

 J \ dx dx J ] dw 



-j 

 dx 



where the integral relative to dcr is supposed to extend over the 

 whole surface, and dx to be the increment of x corresponding to 

 the increment dw. 



In precisely the same way we have 



, 



dw dy ' 



i Cj j frrndU" J7 ,dU'\ f, ds ~dU 

 and j tody (V -^ - F _) = -J^^ F^ ; 



therefore, the sum of all the double integrals in the expression 

 before given will be obtained by adding together the three parts 

 just found; we shall thus have 



~ + d & +} = - [ 

 dx dw dy dw dz dw} j dw 



- r 

 Hence, the integral 



where F and -7- represent the values at the surface of the body. 



dV dU dVdU dVdU 



by using the characteristic 8 in order to abridge the expression, 

 becomes 



i-t- j dxdydz VSV. 

 Since the value of the integral just given remains unchanged 



