. E. B. Wilson — Divergence and Curl. 217 



The value of (5) varies with different orientations of da, from 

 zero when da is perpendicular to ti to its maximum value 

 2co | da | when da is parallel to n. Hence if da be that 

 direction for which the integral is a maximum, 



da 



2wn = 



da. da 



fv.dr. (6) 



This is precisely the definition stated in § 1 for the curve of 

 the function V. It has been obtained on the assumption that 

 ]Tis merely the result of angular rotation. 



The question of whether (6) may be extended to the general 

 case of fluid velocity is not so easy to answer directly. It is 

 well known that if a small portion of the fluid surrounding 

 any point P be selected, this portion is moving at any 

 instant 1° with a velocity of translation which cannot affect 

 the integral of the total velocity about a closed curve, 2° with 

 a motion of rotation which gives precisely the result contained 

 in (5) or (6), and 3° with a motion of dilatation or strain. 

 The influence of this latter on the curvilinear integral is not 

 obvious ; for the magnitude of the displacement due to the 

 strain is of the same order of infinitesimals as that due to the 

 rotation. Nevertheless the strain is taking place symmetrically 

 about the point P, and it is therefore plausible that the total 

 contribution to the integral (5) taken around a closed curve 

 surrounding P should be zero. 



If this plausibility be taken as giving the true state, the 

 relation (6) stands and the general definition may be given, 

 namely : The curl of a vector function V is 



o 



where the integral is taken around a path in that plane which 

 renders its value, relative to the area of the path, a maximum. 

 It is an obvious corollary that the value o'f the expression 

 taken about a path in any other plane is equal to this its 

 maximum value multiplied by the cosine of the angle 

 between the two planes. Furthermore, if any surface be 

 divided into infinitesimal areas, 



curl V.da = ./' V.dv. 



o 



If these expressions be summed up all over the surface, the 

 line integrals will cancel themselves out for all interior 

 lines of division, and hence 



ff curl V.da =/ V.dr, (8) 



S o 



