36 IRROTATIONAL MOTION. [CHAP. III. 



Expressing this statement analytically we have, by (5), 



JJ2 (Zf + mi; -I- ?i?) dS = j(udx+vdy+wdz) ......... (6), 



or, substituting the values of f, 77, f from Art. 38, 



[(( 



M 



JJ { 



-, (dw dv\ , (du dw\ , fdv du 



-7 --- r) + wl [j--^-J+ n [j -j- 



\dy dzj \dz dxj \dx dy 



wdz) ............. (7) ; 



where the double-integral is taken over the surface, and the 

 single-integral along the bounding curve. In these formula 

 the quantities I, m, n are the direction-cosines of the normal 

 drawn always on one side of the surface, which we may term the 

 positive side ; the direction of integration in the second member 

 is then that in which a man walking on the surface, on the 

 positive side of it, and close to the edge, must proceed so as to 

 have the surface always on his left hand. 



The theorem (6) or (7) may evidently be extended to a surface 

 whose boundary consists of two or more closed curves, provided 

 the integration in the second member be taken round each of 

 these in the proper direction, according to the rule just given. 



Thus, if the surface-integral in (6) extend over the shaded portion 

 of the annexed figure, the directions in which the circulations 

 in the several parts of the boundary are to be taken are shewn by 

 the arrows, the positive side of the surface being that which 

 faces the reader. 



The value of the surface-integral taken over a closed surface 

 is zero. 



It should be noticed that (7) is a theorem of pure mathe- 



