115] 



PARTIAL INTEGRATION. 



341 



Now the integral is to be taken between the values of x where the 



edge of the elementary prism cuts into the surface S and where it 



cuts out from the 



surface. If it 



cuts in more than 



once, it will, 



since the surface 



is closed, cut out 



the same number 



of times. Let the 



values of x, at 



the successive 



points of cutting, 



be 



Fig. 122. 



then 



r k being the value of W for x k , and 

 !6) fff^dxdyd^ 



Now let dS^t dS 2 , . . . dS^ n denote the areas of the elements of 

 the surface S cut out by the prism in question at x^, x%, . . . x* n , 

 these all have the same projection on the !FZ- plane, namely dydz. 

 If all these elements are considered positive, and if n be the normal 

 always drawn inward from the surface S toward the space r, at each 

 point of cutting into the surface S, n makes an acute angle with 

 the positive direction of the axis of X, and the projection of dS is 



dy dz = dS cos (nx) 9 



but where the edge cuts out n makes an obtuse angle, with negative 

 cosine, and therefore 



dydz = dS cos (nx) . 



We may accordingly write 



dy dz W = 



cos 



dy dz W B = W 3 cos (n s x) dS 



3 , 



