74 VECTOR ANALYSIS. 



This dyadic is homologous with such as are obtained by varying 

 the values of a, b, c, and only such, unless 6 = 0. 



It is always possible to take three mutually perpendicular vectors 

 for a, /3, and y ; or, if it be preferred, to take such values for these 

 vectors as shall make the term containing c vanish. 



158. The dyadics described in the two last paragraphs may be 

 called shearing dyadics. 



The criterion of a shearer is 



The criterion of a simple shearer is 



{$-I} 2 = 0, 

 The criterion of a complex shearer is 



NOTE. If a dyadic $ is a linear function of a vector p (the term linear being used in 

 the same sense as in No. 105), we may represent the relation by an equation of the form 



< = a/3 y. p + ef t]. p + etc. , 

 or 4> = {afiy + eft + etc. } . p, 



where the expression in the braces may be called a triadic polynomial, and a single term 

 afiy a triad, or the indeterminate product of the three vectors a, |3, 7. We are thus led 

 successively to the consideration of higher orders of indeterminate products of vectors, 

 triads, tetrads, etc., in general polyads, and of polynomials consisting of such terms, 

 triadics, tetradics, etc., in general polyadics. But the development of the subject in 

 this direction lies beyond our present purpose. 

 It may sometimes be convenient to use notations like 



\ M v nA X /*> v 



- - and 3 r 



K /3, y a >A7l 



to represent the conjugate dyadics which, the first as prefactor, and the second as 

 postfactor, change a, /3, y into X, /*, v, respectively. In the notations of the preceding 

 chapter these would be written 



Xa' + /x/3' + vy' and a' 



respectively, a', /3', y' denoting the reciprocals of a, /3, y. If r is a linear function 

 of p, the dyadics which as prefactor and postfactor change p into r may be written 

 respectively 



T j T 

 - and -. 



\P P\ 



If T is any function of p, the dyadics which as prefactor and postfactor change dp into 

 dr may be written respectively 



dr , dr 



j- and -=-:-. 

 \dp dp\ 



In the notation of the following chapter the second of these (when p denotes a posi- 

 tion-vector) would be written VY. The triadic which as prefactor changes dp into 



^- may be written -4, and that which as postfactor changes dp into may be 

 \dp \dp* dp\ 



,72 



written j- . The latter would be written V VT in the notations of the following chapter. 

 dp* I 



