VECTOR ANALYSIS. 57 



s 



120. In like manner, the sum of two dyads may be reduced to a 

 single dyad, if either the antecedents or the consequents are parallel, 

 and only in such cases. 



A sum of three dyads cannot be reduced to a single dyad, unless 

 either their antecedents or consequents are parallel, or both ante- 

 cedents and consequents are (separately) complanar. In the first case 

 the reduction can always be made, in the second, occasionally. 



121. Def. A dyadic which cannot be reduced to the sum of less 

 than three dyads will be called complete. 



A dyadic which can be reduced to the sum of two dyads will be 

 called planar. When the plane of the antecedents coincides with 

 that of the consequents, the dyadic will be called uniplanar. These 

 planes are invariable for a given dyadic, although the dyadic may be 

 so expressed that either the two antecedents or the two consequents 

 may have any desired values (which are not parallel) within their 

 planes. 



A dyadic which can be reduced to a single dyad will be called 

 linear. When the antecedent and consequent are parallel, it will be 

 called unilinear. 



A dyadic is said to have the value zero when all its terms vanish. 



122. If we set 



T = 



and give /> all possible values, <r and T will receive all possible values, 

 if $ is complete. The values of a- and T will be confined each to a 

 plane if < is planar, which planes will coincide if <1? is uniplanar. 

 The values of <r and T will be confined each to a line if 3> is linear, 

 which lines will coincide if < is unilinear. 



123. The products of complete dyadics are complete, of complete 

 and planar dyadics are planar, of, complete and linear dyadics are 

 linear. 



The products of planar dyadics are planar, except that when the 

 plane of the consequents of the first dyadic is perpendicular to the 

 plane of the antecedents of the second dyadic, the product reduces 

 to a linear dyadic. 



The products of linear dyadics are linear, except that when the 

 consequent of the first is perpendicular to the antecedent of the 

 second, the product reduces to zero. 



The products of planar and linear dyadics are linear, except when, 

 the planar preceding, the plane of its consequents is perpendicular to 

 the antecedent of the linear, or, the linear preceding, its consequent is 

 perpendicular to the plane of the antecedents of the planar. In these 

 cases the product is zero. 



All these cases are readily proved, if we set 



o- = 



