24 GENERAL PRELIMINARY RESULTS. . 



To prove this let us consider the triple integral 



The method of integration by parts, reduces this to 



ay 



tlie accents over the quantities indicating, as usual, the values 

 of those quantities at the limits of the integral, which in the 

 present case are on the surface of the body, over whose interior 

 the triple integrals are supposed to extend. 



f rJ TJ" 



Let us now consider the part \dydzV" ^ due to the 



greater values of x. It is easy to see since dw is every where 

 perpendicular to the surface of the solid, that if d<r" be the 

 element of this surface corresponding to dydz, we shall have 



, _ dx , 

 dydz -j- da , 

 dw 



and hence by substitution 



dw dx 

 In like manner it is seen, that in the part 



, dU' 



due to the smaller values of x, we shall have 



dx j , 

 -\- -7 a/(T , 

 aw 



