rNDETEEMIXATE EQUATIONS AND CONGEUENCES. 
317 
in which v«- 2 ? Vn-s? • • • represent the greatest common divisors of the minors of (77.), 
we may replace ||v»-i|| by the matrix 
V„-i 
1 0 
0 v' 
X 
1 , 
Vrt— 1 Vw — 2 
Vi 
Vo 
X 
1 0 
0 u' 
where 
1 0 
Or' 
and 
1 0 
Ow' 
2 3 
denote unit-matrices of the type nxn, the forms of which are 
sufficiently indicated by the sjnnbols themselves. Hence, observing that 
and that 
1 0 
0 u' 
xl^.l, 1,... 
Vn-1 
, 1, 1, ... 
1 0 
Ow' 
1 , 
1 — 2 
V« — 2 Vtt“3 Vo 
we obtain, from (76.), 
, 1, 1, 
» =N X 
or more simply, 
1 0 
X 
'Or' 
i 
«II=!Wlx 
Vn-l 
Vn Vn — 1 
V» V n— 1 
V»~l V« — 2 
Vi 
Vo 
^n— I V^“2 
Vn ^n— 1 
Vi 
Voll 
Vo 
X 
1 0 
Ow' 
X m' 
X 11^11- 
[Vn — 1 Vn— 2 
It has, however, still to be shown that Vn- 2 ? Va- 3 ? • • • which have been defined with 
reference to the matrix (77.) are the greatest common divisors of the successive systems 
of minors of jlajl. These greatest common divisors are the same for the. given matrix ||a|| 
and for the matrix 
Vn Vn— 1 
Vill 
Vn-l’ Vn- 2 ’ ■ 
■ ■ Voll 
unit-matrices; consequently Vn-i dirides every first minor of 
Vn Vn-l 
Vl 
7 7 
Vn-l Vn— 2 
Vo 
, and. 
in particular, it dirides x X X ■ . • X i- e. divides 
Vn 
Vft— 1 Vn — 3 V«— 4 Vo Vn — 1 V/i— 2 V/i — 
Again, Vn-n Vn- 2 . • • • Vn Vo? which are the determinant and greatest common divisors 
of the minors of (77.), are also the determinant and greatest common divisors of the 
minors of the matrix 
Vn— 1 Vn — 2 Vl 
(78.) 
Vn-2 Vn-3’ 
so that if w — 2, v* dirides every minor of order s in (78.), and, consequently, the minor 
X X X — X ^ ; or divides It thus appears that in the series of 
V» V»-2 V»-3 Va V«-i V« 
numbers 
V n Vra— 1 V 2 Vl 
Vn-l Vn -2 Vl Vo 
each term is divisible by that which comes after it. Every product of s terms of that 
series is therefore dirisible by the product X x . . . X — = V* ; or, which is the 
V»-i Vs -2 Vo 
same thing, Vs is the greatest common divisor of the minors of order s in the reduced 
matrix (78.), and therefore in the given matrix ||( 2 ||. 
MDCCCLXI. 2 X 
