VECTOR ANALYSIS. 53 



106. An expression of the form 



a X . p + ft M p + etc. 

 evidently represents a linear function of p, and may be conveniently 



written in the form 



{a\+/3iu.+etc.}.p. 



The expression />.aX + p./3/z+etc., 



or p 



also represents a linear function of p, which is, in general, different 

 from the preceding, and will be called its conjugate. 



107. Def. An expression of the form aX or /3/j. will be called a 

 dyad. An expression consisting of any number of dyads united by 

 the signs + or will be called a dyadic binomial, trinomial, etc., 

 as the case may be, or more briefly, a dyadic. The latter term will 

 be used so as to include the case of a single dyad. When we desire 

 to express a dyadic by a single letter, the Greek capitals will be used, 

 except such as are like the Roman, and also A and 2. The letter I 

 will also be used to represent a certain dyadic, to be mentioned 

 hereafter. 



Since any linear vector function may be expressed by means of a 

 dyadic (as we shall see more particularly hereafter, see No. 110), the 

 study of such functions, which is evidently of primary importance in 

 the theory of vectors, may be reduced to that of dyadics. 



108. Def. Any two dyadics <3> and are equal, 



when <.p = .p for all values of p, 

 or, when p& = p& for all values of p, 

 or, when o-/l > ./o = o-.>E r ./o for all values of <r and of p. 



The third condition is easily shown to be equivalent both to the first 

 and to the second. The three conditions are therefore equivalent. 



It follows that $ = , if 3>.p = .p, or p& = p&, for three non- 

 complanar values of p. 



109. Def. We shall call the vector &p the (direct) product of 

 $ and p, the vector p& the (direct) product of p and 3>, and the 

 scalar <r&.p the (direct) product of <r, <3?, and p. 



In the combination <l?.p, we shall say that 3> is used as a pref actor, 

 in the combination p.<fr, as a postf actor. 



110. If r is any linear function of p, and for p = i, p=j, p = k, the 

 values of r are respectively a, /3, and y, we may set 



r 



and also 



T 



Therefore, any linear function may be expressed by a dyadic as 

 prefactor and also by a dyadic as postfactor. 



