concerning the contents of Pohjf/om and Polyhedrons. 337 



from the other. Each of the lines in cither one of the matrices 

 so reduced in width as aforesaid being then miiltiphed by each 

 hne of the other, and the results of the multiplication arranged 

 as a square matrix and bordered with the two respective sets of 

 columns cut off arranged symmetrically (the one set parallel to 

 the new columns, the other set parallel to the new lines), the 

 complete determinant represented by the new matrix so bordered 

 (abstraction made of the algebraical sign) will be the product of 

 the two original determinants." 

 ah a.^ 

 c d 7 

 lowing forms :— 



aa + b^; ay + bS 

 cx + dj3; cy-\-dS, 

 01- act; ay, b 2; 3; a; A* 



CO.; cy; d or 2; 2; c; d 

 /?; 8; a; ^ ; Oj 



7J 8; 0; 0. 

 And in general for two matrices of ji^ terms each, this rule of 

 multiplication will give(/i + l) distinct forms representing their 

 products. 



Thus ":'^y X " ^ may be put imder any one of the three fol- 



Thus, as a further example, 



a b c a. ^ <y 



d V d X «' /3' j' 



a" b" c" «" ^" 7"; 



besides, the first and last form will be reprcsentable by the two 

 intermediate forms 



"au + b/S a<x' + bl3' 



a'cc + b'/3 a'ct' + b'^' 



a"cc + b"l3 a"c<.' + h"/5' 



c 



ac<." + b/3" 

 a'a." + b'fi" 

 a"c<" + b"/3" 



y" 



' "1 

 dj 



c" j^ 



J 



and 



+ . 



7; 



a"u>', 



7'; 





y; 



b"; 

 0; 

 0; 



1 



♦ Any quantities might be substituted instead of 2 in the places occupied 

 by the figure in the above determinant, as such terms do not influence the 

 result ; this figure is probably, however, the proper quantity arising from 

 tiic application of the rule, liecause (as all who have calculated with deter- 

 minants arc aware) the value of the detcnninant represented by a matrix 

 of no places is not zero but unity. 



Pliil. May. S. 4. Vol. 4. No. 2G. Nov. 1852. Z 



