VECTOR ANALYSIS. 73 



Therefore, since {aaa'+p}.a = aa = &a, 



it follows (by No. 108) that 



156. It will be sufficient to indicate (without demonstration) the 

 forms of dyadics which belong to the particular cases which have 

 been passed over in the preceding paragraph, so far as they present 

 any notable peculiarities. 



If n = 1 (page 72), the dyadic may be reduced to the form 



where a, /3, y are three non-complanar vectors, a, f, y' their reci- 

 procals, and a, b, c positive or negative scalars. The effect of this as an 

 operator, will be evident if we resolve it into the three homologous 

 factors 



The displacement due to the last factor may be called a simple shear. 

 It consists (when the dyadic is used as prefactor) of a motion parallel 

 to /3, and proportioned to the distance from the a-/3 plane. This 

 factor may be called a shearer. 



This dyadic is homologous with such as are obtained by varying 

 the values of a, b, c, and only with such, when the values of a and b 

 are different, and that of c other than zero. 



157. If the planar <3? al (page 71) has perpendicular planes, there 

 may be another value of a, of the same sign as | $ |, which will give a 

 planar which has not perpendicular planes. When this is not the 

 case, the dyadic may always be reduced to the form 



where a, /3, y are three non-complanar vectors, a', ft ', y', their reci- 

 procals, and a, 6, c, positive or negative scalars. This may be resolved 

 into the homologous factors 



al and I + b{a/3' + /3y'} + cay. 



The displacement due to the last factor may be called a complex shear. 

 It consists (when the dyadic is used as prefactor) of a motion parallel 

 to a which is proportional to the distance from the a-y plane, together 

 with a motion parallel to b/3 + ca which is proportional to the distance 

 from the a-/3 plane. This factor may be called a complex shearer. 



