INDETEEMINATE EQUATIONS AND CONGEUENCES. 
307 
Then, if ||m|| represent any unit-matrix of the type wxw, and X„ Ag, . . . A„ absolutely inde- 
terminate integers, the complete solution of the problem is contained in the formula 
Hx||7>,y+^AII 
i—1, 2, 3 ... w > (50.) 
j—1, 2, 3. . . w+1 
For if 11^^11 be any one of the matrices contained in that formula, it is readily seen that 
C 
c 
X 
7 
— CiCj-f CaCa-f- . . — 1. 
= 1, lyi is included in the 
Conversely, if |j.rj| be a matrix satisfying the equation 
formula (50.). To show this, we observe that the complete solution of equation (48.) is 
contained in the formula 
2, ...w + 1, ...... (51.) 
in which |^j| is any set of fundamental solutions of the equation 
Ci^i + C2;y2+ • • • • +C„+,y„+i = 0, (52. ) 
and A,, A.^, . . A„ are indeterminate integers. The complete solution of the same equa- 
tion (48.) is therefore supplied by the determinants of the matrix |!y;,>H-AiCJ. For those 
determinants may be represented by the formula 
2, 3, . . .w+1, 
C 
in which symbolizes a first minor of the determinant 
rCl 
, SO that 
d 
[b 
But the numbers [/, 1], [/, 2], . . . [?, 71+ 1] satisfy (52.) for every value of i; and, since 
Cl 
= 1, the determinants of the matrix 
7\ 
7 = 1, 2, 3, 
2, 3, 
1 
+ 1 / 
(53.) 
are the numbers C,, C^, . . . C„+i, and are therefore relatively prime. It follows from 
this that (53.) represents a set of fundamental solutions of (52.) ; i. e. that the com- 
plete solution of (48.) is represented by the determinants of ilTi-.^+AjCJ. If then ||^|| be 
C . . 
a matrix satisfying the equation =1, since the determinants of ||^|| evidently satisfy (48.), 
cc 
values can be assigned to Aj, Ag, .. . A„ which shall verify the equation 
whence it follows that 
IMNH|x|iy;,,'+Aiq|, 
||m}| denoting a unit-matrix, i. e. |[.r|| is one of the matrices included in the formula (50.). 
The result incidentally obtained in the foregoing analysis, that the complete solution 
