Fundamental Theorem of Determinants. 143 



square there are n^ terms. Now let n be broken up in any given 

 manner into two parts ^ and q^ so that p + qzsn. Let 1°, one 

 of the two given squares be divided in a given definite manner 

 into two parts, one containing p of the n given lines, and the 

 other part q of the same ; and 2°, let the other of the two given 

 squares be divided in every possible way into two parts, consisting 

 of q and p lines respectively, so that on tacking on the part con- 

 taining q lines of the second square to the part containing jo lines 

 of the first square, and the part containing p lines of the second 

 square to the part containing q of the first, we get back a new 

 couple of squares, each denoting a determinant difi'erent from 

 the two given determinants ; the number of such new couples 

 will evidently be 



n.{n--l) . . . (n—p + l) , 

 1.2 ... p ' 

 and my theorem is, that the product of the given couple of deter- 

 minants is equal to the sum of the products [affected with the proper 

 algebraical sign) of each of the new couples formed as above de- 

 scribed. Analytically the theorem may be stated as follows. 

 Let 



r«l «2 • • • «« I /«1 «2 • • • «n 1 



\b, b,...bj L^i /e,...^J' 



according to the notation heretofore employed by me in the pre- 

 ceding Numbers of this Magazine, denote any two common de- 

 terminants, each of the nth order, and let the numbers 6^, 6ci*»*Bn 

 be disjunctively equal to the numbers 1, 2, ... n andjo + g' = ?z; 

 then will 



\b, b^...bnS i^, )e,.../3j 



_^ r«i «2 ... «n "\^r«i «2 ••• ^«1 

 ~\b^ bc^^.bp ^Op+i ffep+2'*»/3en} Xffdi ^e^^.^ep bp+i bp+2*''bn J 



The general term under the sign of summation may be repre- 

 sented by aid of the disjunctive equations 



(pi </>2...^^ = l, 2,...n 



i^i irc^...ylrn=:l, 2,...n, 

 under the form of 



(a^,.b^ X a^^.b^ x ... a^pbp) {a^^_^^.bp+, x a^^_^^.bp+2 x ...a^^ X) 

 ><{^<Pp+v^9p^y^a^^^^.^9p^^x,.M^^.^eJ{a4,^.0g^Xa^,.l3e,x.^^^ 



1st. AVhen ^^ (j><.2 • - • 0,='«/^i '^2 • • • '^r, it will readily be 

 seen, that for given values of <)E)j, <j5)2 . . . <^r; the product of the 



