180 PEOCEEDINGS OF THE AMERICAN ACADEMY. 



Let US return to the discussion of the complete definition of the vec- 

 tor m. All that we have hitherto said of this vector is comprised in 

 the statement, 



Oxm = E + H. 



It is evident that this equation does not completely define m, for in 

 general if m' is a vector satisfying the equation 



Oxm' = E + H, 



we may superpose upon the field of the vector m' the field of another 

 vector m" for which 



Oxm" = 0. 



Then if m = m' + m", we also have 



Oxm = E + H. 

 Suppose now 22 that m'' be so chosen that at every point in the field 



Om" = - Om'. 



Then m satisfies the two equations, 



Oxm = E + H, 

 Om = 0. (81) 



We may, therefore, without in any way modifying what has pre- 

 ceded, complete the definition of m by the equation (SI). This equa- 

 tion combined with (67) gives the well-known expression, 



va-f^ = 0, (82) 



or diva-f-|^ = 0.23 



c di 



Now by equations (53) and (69), 



Oxm = O (Om) - O'm = q, 

 or by (81) 



O'm = - q. (83) 



^^ This is not offered as a rigorous proof, for wc have assumed that a field 

 can be olioscn with pre-dcteriuincd vahies of 0°i" ^^d Oxni". 

 *' Abraham-Foppl, II, equation (30). 



