VECTOR ANALYSIS. 55 



as is at once evident, if we suppose <I> to be expanded in terms of 

 ii, ij, etc. 



116. Def. The (direct) product of two dyads (indicated by a dot) 

 is the dyad formed of the first and last of the four factors, multiplied 

 by the direct product of the second and third. That is, 



The (direct) product of two dyadics is the sum of all the products 

 formed by prefixing a term of the first dyadic to a term of the second. 

 Since the direct product of one dyadic with another is a dyadic, it 

 may be multiplied in the same way by a third, and so on indefinitely. 

 This kind of multiplication is evidently associative, as well as dis- 

 tributive. The same is true of the direct product of a series of factors 

 of which the first and the last are either dyadics or vectors, and the 

 other factors are dyadics. Thus the values of the expressions 



will not be affected by any insertion of parentheses. But this kind of 

 multiplication is not commutative, except in the case of the direct 

 product of two vectors. 



117. Def. The expressions 3>x/o and /ox3? represent dyadics which 

 we shall call the skew products of 3? and p. If 



3? = aX -f /3/z + etc., 



these skew products are defined by the equations 



<l>X/3 = a \ x/o + /5 fi x/o + etc., 



It is evident that 



p*{3>&}, 



a.{3>X/o} = 

 }. 

 We may therefore write without ambiguity 



This may be expressed a little more generally by saying that the 

 associative principle enunciated in No. 116 may be extended to cases 

 in which the initial or final vectors are connected with the other 

 factors by the sign of skew multiplication. 

 Moreover, 



a.pX& = [aXp]& and 3>x/o.a = < 

 These expressions evidently represent vectors. So 



These expressions represent dyadics. The braces cannot be omitted 

 without ambiguity. 



