33] DEFINITE INTEGRALS. 61 



The portion of the integral due to one such prism is 

 dydz^dx. 



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 



1 ) 2 



then \^-dx= TF 2 - W.+ W.-W,... + W zn - 



J ox 



Wk being the value of W for #&, and 



-= dxdydz 



ox 



-InW W-4-W W 4-W -W 1 tlii fly 



[_ "r 2 i TP <f ' 4 ' ' 3 i ' ' 2W '' 271 ij M''/W"2. 



Now let c?^, c?>S^ 2 ... dS 2n denote the areas of the elements of 

 the surface S cut out by the prism in question at a? l5 % 2 , ... x m 

 these all have the same projection on the F^-plane, namely dydz. 

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

 normal always drawn inward from the surface 8 toward the space 

 T, 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 



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

 negative cosine, and therefore 



dydz= d8 cos (nx). 

 We may accordingly write 



dydzW l = W l cos^i^dSj^, 

 dydzW?. = W 2 cos(n 2 x)dS 2 , 



- dydzW m = W m cos 



