IXDETEEmNATE EQUATIONS AND CONOEUENCES. 
303 
by adding the horizontal row 
^r, I! 25 • • • • 714-1 
to the matrix 
MX(w+w) 
X 
equations of the form 
, are identically equal to zero. We thus obtain XX 
m 
n+ 1 
2;=r "p, iw-i, /^25 •• •iM'm-i]^v,i=05 • (31.) 
^ 1 , [m. 2 , .. . representing any combination of m—1 of the numbers 1, 2, 3, . . . 
In connexion with these equations, consider also the similarly formed system, 
1=^2,... (32.) 
This system, which is in appearance redundant (containing X X equations, and 
only m-\-n indeterminates), is in reality defective, and is equivalent to m independent 
equations. For if A’a, ...i’^) be one of the given numbers C which is not equal to 
zero, the partial system of m equations 
(?, ]t\, - 1 }+,, . . . K)yi=^ 
2, 3, ... w 
is certainly an independent system, because the determinant of the coefficients of 
yt,, • ••kmT, and is therefore different from zero. Again, every equa- 
tion of (32.) which is not already comprised in (33.), may be obtained by linearly com- 
bining the equations of that partial system. To verify this assertion, let 
^p5 f^l5 ^25 • • • i~ b 
/’=!, 2, 3, ... 71 
be the system of n equations obtained by attributing to r the n values of which it is 
susceptible in any one of the equations (31.). Eliminating from this system those n — 1 
determinants p, f/J^, ... in which i has a value not included in a set of m-\-l 
numbers v^, v^, . . . arbitrarily selected from the series 1, 2, 3, . . . we obtain a 
relation, which may be expressed in the form 
2:':r^'(— i)'pi,fA„i«'2, ...V, ]=0, . . (35.) 
. mnX^ 
representmg equations, since the sets 
y-'ii ^^25 • . . y^m-\i 
may respectively denote any sets of m —1 and m-f-l numbers taken from the series 
1, 2, . . . m-\-n. Since (35.) is of the form ff>(X„ Xj, . . . X;^)=0, we may at once infer 
the corresponding relation, 
^ 1=1 ( 1) (*'i5 f^l5 f^25 • • • f^m-l) X (t'l, *'25 • . . *'i-i5 *'i4-15 • • • *^m + l) b, . . (36.) 
by means of which any one of the equations (32.) may be deduced from the equations of 
the partial system (33.). Thus, if we multiply the equations of that system taken in 
2 T 2 
