ME. H. J. S. SMITH ON SYSTEMS OF LINEAE 
QOO 
mJ 
or, since 
Vff— 1 
is divisible by P, and 
^ V«-i 
V^+ k 
V‘+4-i 
X 
_ • • • ^ _ 
\S+k^i Vs—J 
by P‘+', 
[P", P"-' Vl, P" ' V 2 , • • • . P Vn-l, Vn] 
= P,XPsXP3 . . . XP.-.XP"'*^' 
=n:::.P, 
But 5i=PiXQiXRiX . • • ; and consequently the greatest common divisor of the deter- 
minants of (83.) is 5, X^aX ••• X^„ or S. Similarly, the greatest divisor of (84.) is 
diXdiX . . ■ X d„ or d. The necessary and sufficient condition for the resolubility of 
the proposed system of congruences is therefore contained in the formula 
d='b. 
It should, however, be observed that, since divides (art. 16), d^ divides and 
-L/s-l Vs-l 
therefore the equation 
d^h 
involves the coexistence of the n equations 
di=\, d^=\, . . . d„:=K (86.) 
To investigate the number of solutions of the system (82.), supposed to be resoluble, 
let |ja|| and ||/3|| be two unit-matrices satisfying the equation 
|x||A||xi|i31|= 
also let 
Vl 
Vo 
. . (87.) 
7=1, 2, 3, ... J 
1 Aj J n-|- 1 ~1~ 2 A 2 , n + 1 I • • • ^i, n -^n, n+ 1 ) 
7=1, 2, 3, . . . w. j 
Then it is evident that the proposed system of congruences is precisely equivalent to the 
system 
V”- i ±j . mod. M, 
Vn-i ‘ 
7=1, 2, 3, ... n 
in such a manner that the two systems are simultaneously resoluble or irresoluble ; and 
that from any number of incongruous solutions of the one an equal number of incoji- 
gruous solutions of the other is deducible. But the whole number of incongruous solu- 
tions of (88.) is ^iX^ 2 X...X^„=^; 7. e. the number of solutions of the proposed system 
is 
By the use of the unit-matrices l|a|| and \\(i\\ the actual resolution of the proposed 
system is made to depend on the resolution of the n congruences contained in (88.). 
But this method of solving a system of linear congruences, though very symmetrical, 
is perhaps too tedious for the purposes of computation. 
