144 On a certain Fundamental Theorem of Detei'minants. 



third and fourth factors becomes mbstantially identical with the 

 general term of the determinant - s 



and consequently, making the system </>!, <^2 • • • </>p (^^I'i which is 

 the same thing, its equivalent yjr^, yjr^ . /. i^^) go through all its 

 jp values, we get back for the sum of the terms corresponding to 

 the equation 0i </>2 • • ' ^iB='»^i V^2 • • • "^py ^^ product of the 

 determinant 1/ f ^^ i)^l^nJfeoi 



/«i «2 . . . «n\ and ^«' ^2 • • • ««\. 

 U, b, ...bj 1/9, fi^...pj 



2nd. When we have not the equality above supposed between 

 the 0*8 and the -^^s, let 



ffip-h^-^p+k and <j>p+,^^'^p~^'y 



the con'csponding term included under the S will contain the 

 factor 



Now leaving <^i, <A2 • • • </*p^ ^^^ '^p '^2' - ^'^p ^^^^^^^^^^t,!^^ 

 may take a system of values ^/, ^2 • - - ^»^ ^^^ch that '^''*' > ^'^ 



and 



and for all other values of q except p + rjj or p — f, ^'^ = 0^. 



The corresponding new value of the general term so formed 

 by the substitution of the 0' for the 6 series, will be identical 

 with that of the term first spoken of, but will have the contrary 

 algebraical sign, because the 6' arrangement of the figures 

 1, 2, 3 . . .p is deducible by a single interchange from the 6 ar- 

 rangement of the same, the rule for the imposition of the alge- 

 braical sign plus or minus being understood to be, that the term 

 in which 



^Op+i ^ep+2 . . '^9n; ^9, /902 . . . ^e^ 

 enter into the symbolical forms of the respective derived couples 

 of determinants, has the same sign as, or the contrary sign to, 

 that in which 



80 enter, according as an odd or an even number of interchanges 

 is required to transform the an-angement 



Op+x 6p+2 • • • ^„> ^i ^2 • • • ^7, 



