338 Mr, J. J. Sylvester on Staudt's Theorems 



To arrive, for instance, at the latter of these two forms, we 

 have only to wi'ite the two given matrices under the respective 

 forms 



and then apply the ordinary rule of multiplication. So, again, 

 to arrive at the first of the above written two forms, we must 

 write the two given matrices under the respective forms 



a b 



a< b' 



a" b" 





 and proceed as before. 



This rule is interesting as exhibiting, as above shown, a com- 

 plete scale whereby we may descend from the ordinary mode of 

 representing the product of two determinants to the form, also 

 known, where the two original determinants are made to occupy 

 opposite quadrants of a square whose places in one of the re- 

 maining quadi-auts are left vacant, and shows us that under one 

 aspect at least this latter form may be regarded as a matrix bo7'- 

 dered by the two given matrices. 



A second but obvious theorem requiring preliminary notice is 

 the following, viz. that the value of the determinant to the matrix 



^2, 1 j ^1,2 j 



-*«, 1 ) "n, 2 } 



ij 1; 



is the same as the value of the detenninant to the matrix 



A2, I ; A2, 2 j 



1; 



• • <if2, » ^ 1 



1; 



• • A2, n ') l- 



A,„ 



1; 



where in general 



/^„ 7*2, . . . h„ and k-^, k^j, . . . kn being any two perfectly arbitrary 



