54 VECTOR ANALYSIS. 



111. Def. We shall say that a dyadic is multiplied by a scalar, 

 when one of the vectors of each of its component dyads is multiplied 

 by that scalar. It is evidently immaterial to which vector of any 

 dyad the scalar factor is applied. The product of the dyadic 3? and 

 the scalar a may be written either a 3? or 3? a. The minus sign before 

 a dyadic reverses the signs of all its terms. 



112. The sign -j- in a dyadic, or connecting dyadics, may be 

 regarded as expressing addition, since the combination of dyads and 

 dyadics with this sign is subject to the laws of association and 

 commutation. 



113. The combination of vectors in a dyad is evidently distributive. 



TVmt i<4 



We may therefore regard the dyad as a kind of product of the two 

 vectors of which it is formed. Since this kind of product is not 

 commutative, we shall have occasion to distinguish the factors as 

 antecedent and consequent. 



114. Since any vector may be expressed as a sum of i t j, and k with 

 scalar coefficients, every dyadic may be reduced to a sum of the nine 



yS ii> ij, ^ ji, jj, jk, ki, kj, kk, 



with scalar coefficients. Two such sums cannot be equal according to 

 the definitions of No. 108, unless their coefficients are equal each to 

 each. Hence dyadics are equal only when their equality can be 

 deduced from the principle that the operation of forming a dyad is a 

 distributive one. 



On this account, we may regard the dyad as the most general form 

 of product of two vectors. We shall call it the indeterminate product. 

 The complete determination of a single dyad involves five independent 

 scalars, of a dyadic, nine. 



115. It follows from the principles of the last paragraph that if 



2 a/3 = 2 K\, 



then 2x/3=S/txX, 



and 2. / 8=2,.X. 



In other words, the vector and the scalar obtained from a dyadic 

 by insertion of the sign of skew or direct multiplication in each dyad 

 are both independent of the particular form in which the dyadic is 

 expressed. 



We shall write < x and $ g to indicate the vector and the scalar thus 

 obtained. 



3> x = (j.3>. k k&.j)i + (k&.i i&. k)j + (i.3>. j j.3>.i)k, 



>. k, 



