564 [May 7, 



obtained as in the ordinary case of the multiplication of the lines or 

 columns of two determinants inter se. Thus, ex.gr. (a, b, c$x,y t s), 



as also ujy) i g to m ean the same product, viz. 

 ax+by + cz. 



Again, imagine a rectangular (square or oblong) matrix of polar 

 umbra, and that each line thereof is multiplied by the same line of 

 actual quantities, the product of the products so obtained I call a 

 Factorial of the Matrix. I also call the product similarly obtained 

 when the columns of the matrix are substituted for the lines, a Factorial 

 of the same, but distinguish between the two by giving to one the 

 name of a Transverse, the second of a Longitudinal Factorial of the 

 matrix. We are now in a position to enunciate the following remark- 

 able theorem : 



The product of any longitudinal by any transverse factorial of the 

 same polar umbral matrix is identically zero, 

 a b c 



Ex. or. Let 



be a matrix of polar umbrae, but x, y, z and 



def 

 also l t rj actual quantities. Then 



(ax + bv \-cz)(dx+ey+fz) 

 is a transverse factorial, 



a longitudinal factorial of the above matrix, and by the theorem their 

 product should be zero. This is easily verified. 

 The two factorials expanded are respectively 



2 + (ae+bd)xy + (bf+ce)yz + (af+dc)zx, 



in their product the coefficient of 

 ff 2 * 8 =abcad=Q, 

 xyl? abcae + abcbd=Q, 

 ,r 2 4 2 ^ = abfad+ aecad+ dbcad= 0, 

 xy^-ri = abfae + abfbd + aecae + aecbd+ dbcae + dbcdb 

 = aecbd-\-dbcae=aecbdaecbd=Q, 



and so for all the other terms. 



This is the fundamental theorem by aid of which I obtain the 



