Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/70

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 $\Phi$ and the scalar $a$ may be written either $a\Phi$ or $\Phi a.$ The minus sign before a dyadic reverses the signs of all its terms.

112. The sign $+$ 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. That is,

$[\alpha +\beta +{\text{etc.}}][\lambda +\mu +{\text{etc.}}]=\alpha \lambda +\alpha \mu +\beta \lambda +\beta \mu +{\text{etc.}}$

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 $i,j,$ and $k$ with scalar coefficients, every dyadic may be reduced to a sum of the nine dyads

$ii,ij,ik,ji,jj,jk,ki,kj,kk,$

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 vector& 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

$\textstyle \sum \displaystyle \alpha \beta =\textstyle \sum \displaystyle \kappa \lambda ,$

then

$\textstyle \sum \displaystyle \alpha \times \beta =\textstyle \sum \displaystyle \kappa \times \lambda ,$

and

$\textstyle \sum \displaystyle \alpha .\beta =\textstyle \sum \displaystyle \kappa .\lambda .$

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 $\Phi _{\text{X}}$ and $\Phi _{\text{S}}$ to indicate the vector and the scalar thus obtained

$\Phi _{\text{X}}=(j.\Phi .k-k.\Phi .j)i+(k.\Phi .i-i.\Phi .k)j+(i.\Phi .j-j.\Phi .i)k,$ $\Phi _{\text{S}}=i.\Phi .i+j.\Phi .j+k.\Phi .k,$