220 E. B. Wilson — Divergence and Curl. 



ate v T'into its self-conjugate and anti-self-con jugate parts ^ 

 and H. 



v F=i(v V+ v V c )+i(v V- v Vc) = * + o. 

 Then / VMr =/ V .8dr +f dr.*.8dr +f dr.tl.8dr. 



o o o o 



As W is self-conjugate, drSP.Bdr is a perfect differential, and 

 O is anti-self -con jugate, 



drM = i v Fx cfr» = |(vxF)x tfr. 

 x 



Hence 



/ r.dr= V .f Mr+if 8(dr.*.dr)+if(v xV)Xdr.8dr. 



Or /" V.dr = (vxV).da, 



o 



from (5) if da denote the area of the curve around which the 

 integral is taken. This equation shows : First, that so far as 

 concerns this problem the fluid appears to be rotating about 

 every point (as a rigid body to infinitesimals of the first order) 

 with an angular velocity |vXF. Secondly, that the objec- 

 tions raised in the latter part of § 3 are therefore void. Hence 

 thirdly, that the definition offered in (7) is perfectly valid. 

 Fourthly, that the Cartesian expression for the curl is 



.- Tr JbZ bY\ jbX bZ\ _ IbY bX\ 



without the necessity of appealing to the application of (7) to 

 an infinitesimal square. Finally that 



'bZ b Y\ 



JfcuAV.da =JJ v X V.da =£f (^ - -) dydz 



s s S 



+ (c§ - b ^) dzdx + (^ - \j) dxd v 



= fv.dv = f {Xdx + Ydy + Zdz). 



o o 



follows properly from the discussions of § 3. 



The introduction of the dyadic V V has therefore accom- 

 plished the justification definitions stated in § 1, has obtained 

 the Cartesian expressions for divergence and curl, and has led 

 naturally and in a very physical fashion to Gauss's and Stokes's 

 Theorems. Thus the program of this paper has been brought 

 to a close. 



Yale University, New Haven, Connecticut. 



