177-180] Green's Theorem 157 



both sides. In the space left, after the interiors of such closed surfaces have 

 been excluded, the functions u, v, w are continuous. We may accordingly 

 apply Green's Theorem, and obtain 



- 2' f j(lu + mv + nw) dS ...... (98), 



where 2 denotes summation over the closed surfaces by which the original 

 space was limited, and ' denotes summation over the new closed surfaces 

 which surround surfaces of discontinuity of u, v, w. Now 

 corresponding to any element of area dS on a surface of dis- 

 continuity there will be two elements of area of the enclosing 

 surface. Let the direction-cosines of the two normals to dS be 

 l lt m lt H! and 1 2 , m 2 , n 2 , so that Ii = l 2 , m l = m 2) and 



= n 



Let these direction-cosines be those of normals 



n a 



drawn from dS to the two sides of the surface, which we shall 



denote by 1 and 2, and let the values of u, v, w on the two 



sides of the surface of discontinuity at the element dS be 



MI, flu w i an d u 2> v 2 , w 2 . Then clearly the two elements of 



the enclosing surface, which fit against the element dS of FIG. 54. 



the original surface of discontinuity, will contribute to 



S' [ \(lu + mv + nw) dS 



an amount dS [(^ + m a v t + nw^ + (I 2 u 2 + m 2 v 2 + n 2 w 2 )] 

 or {1 (MJ - u 2 ) + m l (X v 2 ) + Wi (w l - w 2 )} dS. 



Thus the whole value of 2)' II (lu +mv + nw) dS may be expressed in 

 the form 



2" H$ (*! - u a ) + m l (v, - v 2 ) + Wj (w l - w. 2 )} dS, 



where the integration is now over the actual surfaces of discontinuity. Thus 

 Green's Theorem becomes 



Mdu dv dw\ 777 

 te+ny+w^y 1 * 



lu + mv + nw)dS 



i (MI - O + m, (v, - v 2 ) + n, (w l - w 2 )} dS ...... (99). 



