64 



It has not been noticed, so far as I am aware that these identities are 

 equivalent to simpler identities pertaining to the operator V, as follows : 



(1)' Sdhdihd2h.V=Vdihd2hSdhV+Vd2hdhSdih\7+VdhdihSd2hV 



(2/ V(Vdhdih.V)=dhSdih^— dihSdhV 



In fact (1) and (2) become these (into q) when applied to the elements of 

 volume and surface just as (3) becomes SdhV=di (into q) when applied to 

 the element of length; 



To identify (1) and (1)^, let h be the vector of the mean point of the par- 

 allelopiped whose edges are dhjdih.djh. The outward vector areas "of the 

 two faces parallel to djh.dah are — VdihdahjVdjhdoh, and the correspond- 

 ing values of q are q+JSdh.V-q, q— iSdhV-q; so that sum of the vector 

 areas into q is — VdihdohSdhV-q- Similarly for the other faces. 



So to identify (2) and (2)', the line elements bounding the parallelogram 

 dh,dih are dh,dih, — dh, — djh, and the corresponding values of q are 

 q+^Sdih^.q, q+oSdh^y.q, q — iSd,h\7.q, q— ^Sdh^Z-qandthe sum dhq is 

 dhSd.hy.q— dihSdhy.q. 



To obtain (1) irom (1)^ divide the given volume into infinitesimal parallel- 

 epipeds by any three systems of surfaces, one of which includes the bound- 

 ary of the volume. In summing the terms (1)^ the introduced interior sur- 

 faces between adjacent elements of volume are gone over twice with the 

 vector areas oppositely directed. These surfaces balance one another, 

 therefore, and may be dropped from the summation, leaving the volume 

 integral equal to the surface integral over the boundary of the volume 

 integral. 



We see also that if any discontinuity in q or its derivatives exists within 

 the given volume that the proper way to overcome this is to surround the 

 discontinuity by surfaces and so exclude the discontinuity. Usually this 

 alterg only the surface over which the surface integral extends without 

 aflFecting the volume integral. 



Similarly (2) is obtained from summation of (2)^ and, as every student of 

 integral calculus is aware, (3) is obtained from dq in a similar manner. 



The sectioxs ok the anchor king. By W. Y. Brown. 



