THE DYADICS IN THREE DIMENSIONS. 407 



With a dyadic is associated a number or extensive quantity 



(F) [AiB,] + [AoM +....+ [AnBn] = ^AiBi] 

 and the product of this with an extensive quantity a is 



(G) [A.Bva] + [A^Bo-a] +....+ [AnBn'o] = X[AiBi-a]. 



Now if we observe the above relations (A) to (G) we will see that a 

 similarity runs throughout which can be made the basis of a symbolic 

 notation. This consists in replacing "LAiBi by a symbolic dyad AB 

 and IfCiDi by a symbolic dyad CD. We then have 



AB = i:AiBi, 



A[BX] = i:Ai[BiX], 

 [XA]B = i:[XAi]Bi, 

 CD = XCiDi, 

 AB-CD = '2Ai[BiCk]Du, 

 CD-AB = 2Ci[DiAk\Bk, 



[AB] = Z[AiBl 

 [AB-a] =sU^•5^•ct]. 



The symbolism consists in each case in omitting the summation sign 

 and the subscripts. Conversely each symbolic expression is equiva- 

 lent to the non-symbolic form obtained by introducing summation 

 signs and attaching subscripts to the proper letters. 



By this notation operations on dyadics appear like operations with 

 simple dyads and the results can be expanded and handled much the 

 same as ordinary extensive quantities. Thus if Ai and Bi are points 

 and a a plane, the last expression can be expanded in the form 



[AB-a] = Z[AiBi-a] = '^{Aia)Bi - S(5.a)^i, 



and if we put 'L{Aia)Bi = (Aa)B, X{Bia)A^ = (Ba)A, we have 



[AB-a] = (Aa.)B - (Ba)A, 



which follows the ordinary Grassmann formula for expansion of a 

 product. 



Two dyadics AB and CD have a double product defined by 



AB:CD = [AC] [BD]. 



The significance and properties of these double products will be dis- 

 cussed later. 



