36 IRROTATIONAL MOTION. [CHAP. Ill 



Hence, integrating round a small closed circuit, 



f(udx + vdy + vrdz) 



= ZI(jd* - zdy) + 7?/(zdx - xdz) + /(xdy - jdx). . .(2). 

 The coefficients of f , ?;, f in this expression are double the pro 

 jections of the area of the circuit on the co-ordinate planes, these 

 projections being reckoned positive or negative according to the 

 direction of the integrations. In order to have a clear under 

 standing on this point, we shall in this book suppose that the 

 axes of co-ordinates form a right-handed system ; thus if the axes 

 of x and y point E. and N. respectively, that of z will point ver 

 tically upwards*. Now let &S be the area of the circuit, and let 

 I, m, n be . the direction-cosines of the normal to &S drawn in the 

 direction which is related to that in which the circulation round 

 the circuit is estimated, in the manner typified by a right-handed 

 screwf. The formula (2) then shews that the circulation in the 

 circuit is given by 



2(J*tiwf+n{&amp;gt;8 ....(3), 



or, twice the product of the area of the circuit into the component 

 angular velocity of the fluid about the normal. 



33. Any finite surface may be divided, by a double series 

 of lines crossing ft, into infinitely small elements. The sum 

 of the circulations round the boundaries of these elements, taken 

 all in the same sense, is equal to the circulation round the origi 

 nal boundary of the surface (supposed for the moment to consist 

 of a single closed curve). For, in the sum in question, the flow 



along each side common to two elements comes in twice, once 

 for each element, but with opposite signs, and therefore disap- 



* Maxwell, Proc. Lond. Math. Soc., t. iii., pp. 279, 280. 



t See Maxwell, Electricity and Magnetism, Oxford, 1873, Art. 23. 



