396 MOORE AND PHILLIPS. 



obey the same laws as the corresponding products of points. These 

 products of matrices expressed in plane coordinates we shall call 

 regressive because each additional factor decreases the dimension of 

 the product. 



If a regressive product vanishes it shows that the planes determine 

 a space of higher dimension than a like number of independent planes 

 determine. For example if 



{a^yb) = 



the four planes a, j3, 7, b intersect in a point as is easily shown by 

 writing the matrix in terms of coordinates. 



4. Mixed products and reduction formulas. If the total 

 number of points in a set of progressive matrices is equal to or less 

 than four, the matrices are multiplied together as already explained. 

 If the total number of planes in a set of regressive matrices is equal 

 to or less than four, they are multiplied together in a similar manner. 

 In both cases the product is associative. If the total number of points 

 or planes in two matrices is greater than four,^ we have not defined 

 the product. In that case we replace each of the matrices by its 

 equivalent in complementary elements. We shall say that points 

 and planes are complementary elements and that lines are comple- 

 mentary to lines. For example, we could replace [AB] by [a/S] where 



[AB] = [a^], 

 [ABC] hx a where 



[ABC] = a, 



etc. The total number of elements (points or planes) in the new 

 matrices will then be less than four and the product can be formed as 

 before. If the total number of elements is equal to four, the product 

 will be the same whether the matrices are expressed in points or planes. 

 If the matrices are of different kinds (one progressive, the other 

 regressive) we express one of them by its complementary form. Thus 

 in every case of the product of two factors there is a definite result 

 that has a meaning. We call this the outer product of the two factors. 

 The product of a line [AB] and a plane 7 is the point of intersection 

 of the line and plane considered as having a certain magnitude. For, 



9 It is assumed here that the number of elements (points or planes) in either 

 factor is not greater than four. 



