260 
PROFESSOR H. J. S. SMITH ON THE ORDERS 
m x xm 2 =-f-l, or — 1, mod 8, and the uneven numbers represented by / are either all 
of the linear forms or else all of the linear forms 8#di3, so that / has the 
character ( — l) -8 ” - . 
/*-! F3-1 ' /2-i F 2_! 
The signification of the symbols ¥, ( — 1) 8 "T, ( — 1) 8 M/', ( — 1) 8 8 "T", which 
occur in the Table, is explained in Arts. 6 and 7. In the next article we shall 
establish an auxiliary proposition which is frequently useful. 
Art. 5. “ There exist pairs of forms <p and equivalent to / and F, and satisfying 
the congruences 
<p + z 2 , 
<I>=/3yQA# 2 -bayAz 2 +aj3£ 2 , (7) 
a/3y=], 
for any proposed modulus V ; but this modulus must be uneven, if either / or F is im- 
properly primitive.” 
In the proof of this proposition we shall employ two lemmas of a very elementary 
character. 
(i) A properly primitive form / represents numbers prime to any given number V ; 
and an improperly primitive form / represents the doubles of numbers prime to any 
given number V. 
Letp be any prime divisor of V, and if / is improperly primitive, let p be an uneven 
prime. If one of the numbers a , a a" is prime to p, let a be prime to p ; then if x is 
prime to p and y and z are divisible by p,f will acquire a value prime top. If a, a', a" 
are all divisible by p, one of the three numbers b, V , b" must be prime to p ; let b be 
prime to p ; then if x is divisible by p , and y and z are prime to p, f will acquire a value 
prime to p. 
If /is improperly primitive andy>=2, we may consider \f instead of / and \a, \ci , \ci' 
instead of a , o', a!'; and we may prove in the same way that \f represents uneven numbers. 
Thus, among the p 3 systems of values which can be attributed to x, y, z for the modulus 
p, there are always some which render/(or \f) prime to p ; there are, therefore, among 
the V 3 systems of values which can be attributed to x, y, z for the modulus V, some 
which render/ (or \f) simultaneously prime to every prime dividing V. 
(ii) If QA is uneven, /represents numbers of both the linear forms 4^-j-l and 4#+3. 
One at least of the principal coefficients of f is uneven, because its discriminant is 
uneven: let then a be uneven, and let a'=X , mod 2, a"=p, mod 2; the substitution 
x=x-\-Xy-\-yjZ will transform /into a form/, in which a x , a\, d[ are all uneven, and in 
which, because the discriminant is uneven, either only one, or else all three, of the coeffi- 
cients b x , b\, b'[ are even. The four numbers a[, a", a x + / +/ + 25, + 2#, + are then 
all uneven ; they are all represented by/, that is by/; but they are not all congruous to 
one another for the modulus 4; therefore /represents numbers of both the linear forms 
4&+1 and 4^+3. 
It follows from these lemmas (i) that if/ is an improperly primitive form, we can find 
