45G PROCEEDINGS OF THE AMERICAN ACADEMY. 



and transforming by (36), where in two dimensions fxrfS vanishes, 

 we obtain the form 



idsxt = — f f{dS'V)xt. (60) 



Equations (59) and (60) can be combined into the operational 

 equation 



fds{) = - ff{dS'V){), (61) 



where the operators may be apphed to f in either inner or outer 

 multiphcation. 



In three dimensions Stokes's theorem states that the line integral 

 of a vector around a curve is equal to the surface integral of the normal 

 component of the curl of the vector over any surface spanning the 

 curve, with proper regard to sign. The ordinary statement is 



ff/s-f = J fdS {curl i)n, 



which in our notation becomes 



Jds't= JJdS'(V't); 

 and may be transformed by (35) into 



fdS't= — j f{dS'V)-t (62) 



In like manner Gauss's theorem states that the integral of the flux 

 of a vector through a closed surface is equal to the integral of the 

 divergence of the vector over the volume inclosed by the surface. 

 Thus, if d^ is the scalar element of volume, 



/ finds = JJJdWtd^. 



In our notation, if dS denotes vector element of volume, this 

 becomes 



//tfS.f)- = ffpBvt = ffp3*V-i. 



which, by transformation by (24) and (32), becomes 



JJdSxt= j JJidS'V)xt (63) 



