324 
ME. H. J. S. SMITH ON SYSTEMS OF LINEAE 
And this equation is impossible, if — 1„. For Is+i— Is_i, because is 
Ug 
divisible by the inequality, I„+i<I„+^, involves, therefore, the inequalities 
In+1 fn-s+1 I 
5=1, 2, 3, ... 
(94.) 
and these, again, imply the corresponding inequalities 
In + l s+l 1 
s^l,2,B,...n+lJ * 
because I„_s+i From (94.) it appears that the value of 
is and from (95.), that the value of is a power of p 
superior to^;'«+i ; ^. e. the equation (93.) is impossible. We thus obtain, as a first con- 
dition for the resolubility of the proposed system (92.), the congruence 
mod. M (96.) 
hen this condition is satisfied, we obtain from (93.), omitting the term p^n+i (because 
I„+ 5^I„+^), and dividing by^®, the equation of condition. 
[pin ^ p^n-l+^, p*n-2+®®, , . . p"®] 
= [^'«, p'n-2+®®, . . . J?”®], 
which leads us (as in the last article) to the simple formula 
d=h. 
This equation, therefore, and the congruence (96.), express the necessary and sufficient 
conditions for the resolubility of the proposed redundant system. 
When these two conditions are simultaneously satisfied, the number of incongruous 
solutions is h. For, if we again consider the proposed system of congruences with 
respect to the modulus p^, and select from it a partial system of n congruences such that 
the determinants of its augmented matrix, which are necessarily divisible by p\ are not 
divisible by any higher power of p, it is readily seen that every set of values of the 
indeterminates x^, x^,... x„, which satisfies the partial system, will also (by virtue of 
the inequality — 1„) satisfy the remaining congruences of the proposed system. 
The number of solutions of the proposed system is therefore the same as that of the 
partial system. And because ^i» the highest power of^ which divides every determinant 
of order n in the augmented matrix of the proposed system is also the highest power of 
p which divides the augmented matrix of the partial system, it follows from the last 
theorem of art. 16, that are the highest powers of^ which divide the 
corresponding orders of determinants in the latter, as well as in the former matrix. The 
number of solutions of the partial system (and consequently of the proposed system), 
considered with respect to the modulus^®, is therefore expressed by the formula 
[pS 
A 
