IXDETEEMINATE EQUATIONS AND CONUEUENCES. 
315 
so on ; i. e. since we can assign to the indeterminates values simultaneously satisfying 
all these congruential conditions, we can give to Ui, Uj, • . . U„ values the greatest 
common di\isor of which is prime to M. Let be the greatest divisor of ||A||, D„_i 
the greatest common dmsor of the first minors of ||A|| ; and let Cj, Cg, . . . C„ be a set of 
simultaneous values of Ui, Ug, . . . U„, ha\ing a greatest common divisor c, which is prime 
to T^— . Since the equations 
A,- iiT] I A; 2^2 I" •• • n + m'^n+m 
i=l, 2, 3, ... « 
are resoluble, it will follow from the condition of resolubility (see art. 11), that the 
determinants of its augmented matrix, and in particular those which contain the column 
C,, Q, . . . C„, are divisible by D„. Let ^XcxD„_i be the greatest common divisor of 
these last determinants; then ^x^xD„_, is divisible by D„, i. e. 0 is divisible by 
It appears from this, that the condition of resolubility^ is satisfied by the system 
iAi+A,_2a’2+ . . . +A,_ 
n+m^' n + m ' 
i=\, 2, 3, . . . 
Q 
c 
that is to say, it is possible to obtain a simultaneous sy^stem of relatively prime values 
for U„ U 2 , . . . U„. 
To apply this principle to the transformation of the matrix |jaj|, let 
dS/n 
duij 
(69.) 
Tlie constituents of the matrix i|[«]j| do not admit of any common divisor ; consequently, 
in the system 
[^5-, i]^i, ij^2, 1 + • • • +[^'L, >]^H, 1 : 1 t"!) ) 
?=1, 2, 3, . . . J’ 
we can assign values to ,, 1 , . • . i, which shall render ,, u.^, 1 , • • • 1 relatively 
prime. Let ||m|| denote a unit-matrix of which the first column is iq 1 , t ; 
and ll^ll a square matrix of which the first column is ,, i, • ■ • n of which the 
remaining constituents are defined by the equations 
"1 
2=1, 2, 3, . . . ^. (71.) 
j— 2, 3, . . . %. j 
Observing that the systems (69.) and (70.) involve the inverse system. 
(K l 22 i, 1 +< 5 ?;, 22«2, 1 + • • • A, , — i,l 
’ \n-\ 
J 
i=l, 2, 3, . . . 71, 
we infer that the matrices ||m|| and ||Z»[| verify the equation 
INIxlNll=iH|x 
Vn — 1 
1 , 1 ,... 
(72.) 
(73.) 
