358 HITCHCOCK. 



dimensions with respect to ordinary dot product. Therefore all 

 properties of such vectors which pertain to dot multiplication alone 

 will go over into properties of polyadics and multiple dot product. 



To say the same thing in another way, the fundamental polyads 

 e p e g - -e s form a normal orthogonal system with respect to multiple 

 dot multiplication, in the sense that the product of two unlike polyads 

 of order K is zero while the product of a polyad by itself is unity. 

 The notation E, for the polyad is convenient but not essential. The 

 analogy with e r , while of much value in a formal sense, is temporary 

 in character, to be employed or laid aside according as we wish to 

 forget or to emphasize the K -tuple nature of the polyads. 



3. The Fundamental Identity of Dot Multiplication. 



Let there now be a set of N K or n polyadics of order K as Ai, A 2 , • • • 

 A n , arbitrarily chosen. Any one of these, as Ai, may be expanded in 

 the manner of (9), thus 



Ai = flijEi + « l2 E 2 H h AinEn. (11) 



Let there also be another polyadic which we may on occasion call 

 A n+ i but which it will be more convenient at present to call M and to 

 expand in the form 



M = J/xEi + il/ 2 E, H h M„E„. (12) 



If we now take the multiple dot product M:A,, remembering the 

 principle of orthogonality pointed out in the last article, we shall have 



M : Ai = Mian + M 2 a a -\ 1- M n a in . (13) 



There are n equations of this form, one for each of the polyadics 

 Ar • -A n . Together with (12) we thus have n + 1 equations linear 

 in the scalars M\---M n . It is true that (12) is a /v-adic equation 

 and is itself equivalent to n scalar equations. But since all the 

 equations (13) are merely scalar equations, it is not hard to see that 

 the determinant 



M , E,, E 2 , •••, E„ 



M:A], an, a r; ,---, r/,„ 



M:A 2 , an, (h-i,- • ■ , a 2n (14) 



M:A„, a, a, a n 2,''', a nn 



must vanish; because when polyadics have to be multiplied by scalars 

 only, all the laws of ordinary algebra are obeyed. 



