32-33] CIRCULATION. 37 



pears from the result. There remain then only the flows along 

 those sides which are parts of the original boundary ; whence the 

 truth of the above statement. 



Expressing this analytically we have, by (3), 



S(udx + vdy + wdz) = 2//(Jf + mrj + n{ ) dS ......... (4), 



or, substituting the values of f, ?;, f from Art. 31, 



j('iidx + vdy + wdz) 



dw dv\ du dw\ dv du 



where the single-integral is taken along the bounding curve, 

 and the double-integral over the surface. In these formulas 

 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 seeead-fnember 

 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 (4) or (5) may evidently be extended to a surface 

 whose boundary consists of two or more closed curves, provided 

 the integration in the first member be taken round each of 

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



Thus, if the surface-integral in (5) 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 



* This theorem is attributed by Maxwell to Stokes, Smith's Prize Examination 

 Papers for 1854. The first published proof appears to have been given by Hankel, 

 Zur allgem. Theorie der Bewegung der Fliissigkeiten, Gottingen, 1861, p. 35. That 

 given above is due to Lord Kelvin, I.e. ante p. 35. See also Thomson and Tait, Natu- 

 ral Philosophy, Art. 190 (j), and Maxwell, Electricity and Magnetism, Art. 24. 



