INDETEEMINATE EQUATIONS AND CONGEUEAtceS. 
305 
■which we have represented by y, is evidently (At, . A:, x c. If, therefore, C, one 
of the given numbers Ci, Cg, ... C^, be a unit, we have only to take C for (At, At .. . A^J, 
and we shall immediately obtain a matrix |jy|| of the type mX{m-\-n) satisfying the 
equation 
i 7 :=iq- 
And similarly might a matrix of the tyq)e nx{m-\-n), satisfying the same equation, be 
written down without calculation. 
Art. 9. The importance of the case, in -svhich m=l, is so great, that we may be 
allowed to point out the identity of the solution obtained by the preceding method 
A^-ith that already given by M. Heemite. Let, then, Cj, Ca, ... C„+, represent the deter- 
minants of a matrix of the tj'pe taken in their natural order [i. e. so taken 
that if the matrix be completed by an additional row of constituents, 
Cl-) ^*2? • • • ^/l+l? 
the value of its determinant Avould be 
C1C1 + C2C2 + C3C3+ . . . Ca+iC„+i). 
We have then to obtain a set of fundamental solutions of the equation 
CiA^i + C2_y2+C3i/3+-- •+C«+ii/H+i = 0 (43.) 
Such a set may always he obtained by the following particular method. Supposing 
that C, is not zero, consider the equations 
0 — C,y,-1-C2y2 
] 
0 = C’l "b C 2 ^2+ ^3^3 
!> (44.) 
b = Cl.yi + t'2^2+C33'3+ • • ■ 
■ • • +C„+iA+i J 
and take a particular solution of each of them, assigning to the last indeterminate in 
each, the least value (zero excepted) of Avhich it is susceptible. If we denote by A* the 
greatest common divisor of C„ C 2 , . . . C\., so that A,=:C„ A,,+, = c, it is evident that the 
A'alue of in the equation 
Ci.yi + C2y2+ • • • — 0 
A* 
will be ^ — ; and if in the same equation we represent the values of //,, . 
the matrix 
/A- by 
A. n r 
k, 2 ? A, 35 • • 
• • A. k) 
, 0, 
0 . . . 
. . 0 
A. 
A3 ’ 
0 . . . 
. . 0 
2 ? ^ 3 , 3 i 
A3 • * • 
. . 0 
A, I ? A, 21 ^ n,Z-) ^ n, 4 
0 
( 45 .) 
