58 VECTOR ANALYSIS. 



and consider the limits within which a- varies, when we give p all 

 possible values. 



The products "^"x/o and yox$ are evidently planar dyadics. 



124. Def. A dyadic <1> is said to be an idemfactor, when 



&.p = p for all values of p, 

 or when /o.4? = /o for all values of p. 



If either of these conditions holds true, 3? must be reducible to the 

 form 



Therefore, both conditions will hold, if either does. All such dyadics 

 are equal, by No. 108. They will be represented by the letter I. 



The direct product of an idemfactor with another dyadic is equal to 

 that dyadic. That is, 



1.$ = *, $.1 = $, 



where <& is any dyadic. 



A dyadic of the form , , Q0 , , , 



act -rpp -ryy, 



in which a', /3', y are the reciprocals of a, /3, y, is an idemfactor. 

 (See No. 38.) A dyadic trinomial cannot be an idemfactor, unless its 

 antecedents and consequents are reciprocals. 



125. If one of the direct products of two dyadics is an idemfactor, 

 the other is also. For, if 3>.ty = I, 



for all values of cr, and <3? is complete ; 



(7.$..3> = o-.<i 

 for all values of or, therefore for all values of 0-.$, and therefore 



Def. In this case, either dyadic is called the reciprocal of the 

 other. 



It is evident that an incomplete dyadic cannot have any (finite) 

 reciprocal. 



Reciprocals of the same dyadic are equal. For if 3> and ty are both 

 reciprocals of ft, $ = $. Q. = ^. 



If two dyadics are reciprocals, the operators formed by using these 

 dyadics as prefactors are inverse, also the operators formed by using 

 them as postfactors. 



126. The reciprocal of any complete dyadic 



a A + ftfji + y v 



is AV+M73 / +"Y> 



where a', /8', y are the reciprocals of a, ft, y, and A', JUL', v are the 

 reciprocals of A, p, v. (See No. 38.) 



