26 GENERAL PRELIMINARY RESULTS, 



when we substitute F in the place of U and reciprocally, it is 

 clear, that it will also be expressed by 



-idffU d ~- I dxdydz mV. 



Hence, if we equate these two expressions of the same quantity, 

 after having changed their signs, we shall have 



Thus the theorem appears to be completely established, what- 

 ever may be the form of the functions U and V. 



In our enunciation of the theorem, we have supposed the 

 differentials of U and V to be finite within the body under 

 consideration, a condition, the necessity of which does not ap- 

 pear explicitly in the demonstration, but, which is understood in 

 the method of integration by parts there employed. 



In order to show more clearly the necessity of this condition, 

 we will now determine the modification which the formula must 

 undergo, when one of the functions, U for example, becomes 

 infinite within the body ; and let us suppose it to do so in one 

 point p only : moreover, infinitely near this point let U be 



sensibly equal to - ; r being the distance between the point p' 



and the element dxdydz. Then if we suppose an infinitely 

 small sphere whose radius is a to be described round p, it is 

 clear that our theorem is applicable to the whole of the body 



exterior to this sphere, and since, BU= & - = within the sphere, 



it is evident, the triple integrals may still be supposed to extend 

 over the whole body, as the greatest error that this supposition 

 can induce, is a quantity of the order a 2 . Moreover, the part of 



r 

 \ 



dcrU~T~ , due to the surface of the small sphere is only an 



infinitely small quantity of the order a; there only remains 



/fjTT 

 foV-j due to this same sur- 



face, which, since we have here 



