464 PROCEEDINGS OF THE AMERICAN ACADEMY. 



The two expressions O^P and 0*p may now be obtained by inserting 

 the cross and dot in OP- Hence 



0-P= -~(l'C+'^^^^hw+ l) = 0. (78) 



Here also O ' P vanishes, since 1 • w = — R. 



As 1 varies with R, the parts of O^P "^ay be separated into one 

 which varies as R~^ and one which varies as R~", namely. 



P = Oxp = - ^, (ixc + ^-^ Ixw) - ^3 Ixw. . (79) 



This may be brought out most clearly by expressing I as 



1= R{w — n), (80) 



w^here n is a unit vector from Q perpendicular to w. 



P = — „ [wxc — nxc + n-c nxw] + „:, nxw. (81) 



Another manner of expressing P is 



P = - -^3 lx[l. (wxc)] - ~3 Ixw (82) 



or 



P = — W3 (Ixwxc)-l — -^3 Ixw. (;83) 



Any of these forms of P shows, what perhaps appears clearest 

 from (82), that the part of P which varies inversely as i? is a singular 

 plane, through the element 1 and cutting the plane of Wxc; for 

 lx[l«(wxc)] is a plane through 1 and the vector l'(wxc) (in wxc), and 

 the inner product of the plane by itself is readily shown to be zero. 



In a similar manner we may calculate OP> ^ dyadic with its first 

 vectors 1 -vectors and its second vectors 2-vectors. The differentiation 

 requires nothing new except <C>C- And by the same reasoning applied 

 to find Ow, it appears that 



Oc = , , = — r, , - (84) 



I'Vfds R (Is 



