VECTOR ANALYSIS. 59 



127. Def. We shall write $" 1 for the reciprocal of any (complete) 

 dyadic <1>, also <1? 2 for $.$, etc., and <~ 2 , for $~ 1 .$~ 1 , etc. It is 

 evident that $~ n is the reciprocal of 3> n , 



128. In the reduction of equations, if we have 



we may cancel the 4> (which is equivalent to multiplying by $~ l ) if 

 $ is a complete dyadic, but not otherwise. The case is the same with 

 such equations as 



To cancel an incomplete dyadic in such cases would be analogous to 

 cancelling a zero factor in algebra. 



129. Def. If in any dyadic we transpose the factors in each term, 

 the dyadic thus formed is said to be conjugate to the first. Thus 



and 



are conjugate to each other. A dyadic of which the value is not 

 altered by such transposition is said to be self-conjugate. The con- 

 jugate of any dyadic < may be written $ c . It is evident that 



p& = 3> c .p and 3?.p = /9.3> c . 



3> c .p and <./> are conjugate functions of p. (See No. 106.) Since 

 {<j> c } 2 = {<!> 2 } c , we may write $c> etc., without ambiguity. 



130. The reciprocal of the product of any number of dyadics is 

 equal to the product of their reciprocals taken in inverse order. Thus 



The conjugate of the product of any number of dyadics is equal to 

 the product of their conjugates taken in inverse order. Thus 



Hence, since 3> .{<Ir 1 } c = { < 3?- 1 . < } c = I, 



{$- 1 }c={^ c }- 1 , 



and we may write ^c 1 without ambiguity. 



131. It is sometimes convenient to be able to express by a dyadic 

 taken in direct multiplication the same operation which would be 

 effected by a given vector (a) in skew multiplication. The dyadic 

 Ixa will answer this purpose. For, by No. 117, 



/ o.{IXa}=/oXa, 



The same is true of the dyadic ax I, which is indeed identical with 

 Ixa, as appears from the equation I.{axl} = {Ixa}.I. 



