WILSON AND LEWIS. — UKLATIVITY. 497 



convenience when the indivi(kial vectors are expressed in terms of 

 coordinate vectors. Thus, 



(/iikiki + (/I'jkik-j 4- "i.<kik:j + f/nkiki 



+ </jikjki + (/■..•..tjk- + a-jkjk.! + (/^ik-jkt 



+ <r:nk:iki + a.-j.jknko + f':«k:ik;i + f/.Tikiiki 



+ f'jiktki + (/-i2k4k-j + f/mkik-j + «44k4k4. 



The product of a \ector a and a dyad be is expressed and defined 



as 



a«bc = a'(bc) = (a«b)c, 



It is a 1-vector along C. Siniihuly ab'C = (ab)*C = a(b'C). The 

 product of a vector into any dyadic follows from the distributive law. 

 The product of two dyads is expressed and defined as follows. 



ab'Cd = (ab)«(cd) = a(b'C)d = (b'C)ad. 



It is another dyad. The product of two dyadics then follows from the 

 distributive law, and is therefore a dyadic. 



Since the dyad product is obtained without implying any relation 

 between the sixteen units k,k^, it is the most general product and com- 

 prises within itself the more special products which we have desig- 

 nated as the inner and outer products and which we may obtain from 

 it by inserting the special sign of multiplication corresponding to these 

 products, thus giving respectively a scalar or a 2-vector. Hence 

 from any dyadic a scalar or a 2-vector may be obtained by converting 

 each dyad into an inner or outer product. This method was employed 

 in computing <0>*P and O^P in § 43 and § 44. 



A dyadic is said to selfconjugate when for all the coefficients 

 (tij = Oji, and anti-selfconjugate when for all the coefficients Oy = — o^j. 

 The latter can have no terms in the main diagonal, and therefore 

 has but six degrees of freedom, whereas the selfconjugate dyadic has 

 ten.^^ Except for sign the anti-selfconjugate dyadic not only deter- 

 mines, but conversely is determined by, a 2-vector of the form 



ai.jki2 + «!3ki.-i + Ouku + ^'23k-2;i + «24k24 + du^: 



34, 



where ai2, . . . are the coefficients of kik^, ... in the expanded form of 

 the dyadic. This 2-vector is one half the 2-vector obtained by insert- 

 ing the sign of outer multiplication in the dyads constituting the dya- 

 dic. 



66 Any dyadic may be written as the sum of two dyadics one of which is 

 selfconjugate, the other anti-selfconjugate. 



