where m. is defined in Equation (314) and L. is the intersection of S_ 

 with the plane z = 0. This theorem can be proved by means of Stokes' 

 theorem as Tuck did, but here let us derive the same result by integration 

 by parts. 



In the first place, we put i = 1 and consider an integral 



'■■ft: - 

 ■If 



) +vd) +w<b ) n,dS 

 x y z 1 



sgnx(u<f> +v<J) +w<J) ) dydz 



If we write the equation of S_ as 



x = f(y,z) 



we have 



Integrating by parts 



3x 



By" (<},) x=f(y,z) = ^y + ^x 8y 



dz vcfyly = dz v 



9 ^x=f 

 3y 



K 97 



= -jdzj (v+v x fj)<Ddy.- 

 = 1 I (v y <t>n 1 -v x c|>n 2 -v(j) x n 2 ) dS 



dy 



'M v(f» x || dy 



By a similar way 



dy 



w(J> z dz = dy 



9( ^x=f A 3x 



dZ X dZ 



dz 



= j dy • w<t> z=Q - JdyJ (w z +w x |*) 4,dz - j dy J wcf, 



n^cbwds - I I (w tbn^-w dm -w<i n„) dS 

 1 II z 1 x 3 x 3 



3x 



x 8z 



dz 



112 



