THE DYADICS IN THREE DIMENSIONS. . 409 



Let Ai, Ai, Az, Ai, be four points, not in a plane, with magnitudes 

 so chosen that 



Ui^2.-l3.l4) = 1. 



Let [^i^2^3] = ai, [A1A2A4] = —as, 



[AiAsAi] = ao, [AoAzAi] = — ai. 



It is then easily seen that 



[aiAi]=l, i = l, 2, 3, 4 (25) 



[aiAj] = 0, i^j. (26) 



Four points Ai and four planes Ui satisfying these equations are said 

 to form a reciprocal system}'^ 



If the points Ai and the planes Cj form a reciprocal system, the 

 dyadic 



/ = Aiaj + Aoa-y + Asaz + ^404 (27) 



is an idemf actor. For equations (25), (26) show that 



lAi = Ai. 



Since any point X can be expressed in the form 



X = XiAi + X2A2 + X3A3 + 0:4^4, 

 it follows that 



IX = xiAi + X2A2 + X2A3 + .r4^4 = X. 



If the antecedents are points lying in a plane or the consequents 

 are planes passing through a point, the dyadic is called singular. 

 Suppose Bi, B2, B3, Bi lie in a plane. Then Pa can be expressed as a 

 linear function of Pi, P2, P3. Hence the dyadic 



5/3 = 5i/3i + £2/32 + ^3^3 + 54/34 



can be written in the form 



Byi = Bai + ^272 + 5373. ' (28) 



12 The reciprocal system <vas fundamental in Gibbs' work on dyadics. See 

 Gibbs-Wilson Vector Analysis. Also the paper by Wilson mentioned in 

 note 3. 



