Mr. CAYLEY, on THE THEORY OF DETERMINANTS. 



77 



Act + A'a'+ . . = K, (7). 



A(i + A'(i'+.. = o, 



Ba + B'a + . . = 0, 



&c. 



The second side of each equation being (o), except for the r"* equation of the »•"' set of 

 equations in the systems. 



Let X, /u> • • • represent the r"", r + l"", ... of the series a, /3, . . . , L, M, . . . the corresponding 

 terms of the series A, B . . . , r being any number less than (w), and consider the determinant 



A, ... L 



(8). 



which may be expressed as a determinant of the »"' order, in the form 



A . . . L, 0.. 



^<'-"...Z,<'-"o 



10 



1 



Multiplying this by the two sides of the equation 



(9). 



(10), 



«, /3.. 



and reducing the result by the equation (O ), and the equations (6), the second side becomes 



K .. 



K 



which is equivalent to 



K . 



m"* I'''' 



M*'""*, >'''■""* 





.("). 



Or we have the equation 

 J .... L 



which in the particular case of r = w, becomes 



A, B . 

 A', S 



(12). 





(13), 



(11), 



