38 40.] CIRCULATION. 35 



cuit. Again f(YdZ ZdY) is twice the area of the projection of 

 the circuit on the plane yz, and therefore equal to 2ldS, where dS 

 is the area of the circuit, and I, m, n the direction-cosines of the 

 normal to its plane. The coefficients of rj and f give in the 

 same way, on integration, %mdS and 2ndS, respectively. Hence, 

 finally, the circulation round the circuit is 



(5), 



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

 angular velocity of the fluid about the normal to its plane. 



We have here tacitly made the convention that the direction 

 of the normal to which I, m, n refer, and the direction in whicli 

 the circulation in the circuit is estimated, are related in the same 

 manner as the directions of advance and rotation in a right-handed 

 screw *. 



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

 straight lines crossing it, into an infinite number of infinitely small 

 elements. The sum of the circulations round the boundaries of 

 these elements, taken all in the same sense, is equal to the circu 

 lation round the original 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 



Fig. 2. 



comes in twice, once for each element, but with opposite signs, 

 and therefore disappears from the result. There remain then only 

 the flows along those sides which are parts of the original bound 

 ary; whence the truth of the above statement. 



* See Maxwell, Electricity and Magnetism, Art. 23. 



32 



