Adding the above equations and making use of relations of continuity and 

 irrotationality, 



v+w=-u, v = u . w=u 

 y z x x y x z 



together with Equation (319), we obtain 

 I 



1 = J (u x n l +U y n 2 +U z I1 3 ) * dS " 1 n l* Wd£ 

 JJ L Q 



= JJ fH * dS " J L n l (f)wds 



= -I I m^dS - I n (fiwds 



A similar relation is obtained when i = 2, 



T 2 = 



" jj m 2* ds " J L n ; 



())wds 



In the case of i = 3, we write 

 I 



■ -JJ- 



d> +vd> +wd> )dxdy 

 x y z 



We get similarly, as before 



4>dxdy + I I u<J) dydz 



-jjudyixdy = - Jdyj (u f£ -u^ ff ) 



= {f( U x +U Z f)* dxd y + JJ U ' 

 = "J J (U x n 3- U z n l ) * dS + Jj u Vl 

 J vcj> y dxdy = - J dxj (v |± -v<j, z |^) dy 



= JJ (V v z ly ) <(>dxdy + JJ v * z dxd 

 " -j| (v y n 3- V Z n 2 ) * dS + {[ V<i, z n 2 



113 



