426 MOORE AND PHILLIPS. 



functions of other extensive quantities; the double product of the 

 dyadics in the new form will be equal to that in the old. The double 

 product is thus independent of the form in which the dyadics are 

 expressed. Hence it is a covariant of the two dyadics. 



The geometrical interpretation of the double product [AC] [BD] 

 depends on whether [BD], that is [BiDi^, is a progressive or a regressive 

 product. Suppose [BD] is progressive and X a space complementary 

 to [BD]. Express X as a product of planes ^», 



Divide the planes into two sets. Call the product of the planes in 

 one set aj and the product of the planes in the other set jSj and arrange 

 the planes in the sets so that 



X= [ai^l 



If the sets are so chosen that /Sj is complementary to D, the reduction 

 formulas (§4) give 



[DX] = llaiWi), 



the summation being for all combinations of the planes, ^1^2. • Im 

 in sets aj, /S^. Therefore 



[AC] [BDX] = [AC] [B-DX] = UC]2(5ai) (Z)/30. 



This result can be written 



[AC] [BDX] = nA{Bai)C{D^i)]. 



This sJiows that if aj is transformed by AB and jSj by CD and if the trans- 

 forms are then joined, [AC] [BDX] tcill be a linear function of the joins. 

 This is true in whatever way X is expressed as a product of planes. 



If [BD] is regressive we proceed as before except that X is expressed 

 as a product of points instead of planes. If B and D are of comple- 

 mentary dimensions, either points or planes may be used. 



Suppose, for example, 



AB = rAiBi, CD = ^CiDi 



Ax, Bx, Ci, Di being points. Let L be any line and ^1, ^2, two planes 

 through it. Transform ^1, by AB and ^2 by CD. The result is two 

 points whose join is a line p. Transform ^^hy AB and ^1 by CD. Let 

 the join of the corresponding points be q. Then [AC] [BD] transforms 



