288 
PROFESSOR H. J. S. SMITH ON THE ORDERS 
tj being a coefficient of which the value is 1, b i, or as shown in the following 
Table. 
(i) (/) properly primitive. 
12=1, mod 2. 
12=2, mod 4. 
12=4, mod 8. 
12=0, mod 8. 
A=l, mod 2. 
1 
2 
1 
i — i 
l\^ 
r— 1 
+ 
CO 
1 1 
h|tH 
>Hh 
i — i 
00 
+ 
T 
1 i 
A =s-2, mod 4. 
1 
2 
1 
* 
i 
A =4, mod 8. 
JL 
1 
2 
1 
2 
1 
2 
| AsO, mod 8. 
1 
4 
1 
4 
1 
4 
1 
4 
(ii) (f) improperly primitive. 
A=2, mod 4. 
i 
3 
A=0, mod 4. 
To establish the equation (45), we consider separately the different primes dividing V. 
And first let us take an uneven prime d, dividing A but not 12. Of the S 3 systems of 
values of x, y, z, mod S, 
S 3 x (i — |) X i [i + ( — i] systems 
give to xz — y 1 a value prime to d, and satisfying the equation (Lemma i, Cor.) 
(xz—y^X /OF\ 
(— )=(t> 
Secondly, let us consider an uneven prime a dividing 12 but not A ; and let u' be the 
highest power of a> dividing 12. Of the <u 3i+3 systems of values of x, y, z, mod cJ + \ 
“"'X^X (l-l) X (l-^) systems, 
in which x, y. z are not simultaneously divisible by a, render xz—y 2 divisible by u\ and 
also render the quotient XS ^ prime to u (Lemma iii. Cor. 1). 
Thirdly, let us consider an uneven prime r dividing both A and 12, and let r' be the 
highest power of r dividing O. Of the r 3i+3 systems of values of x, y , z, modr' +I , 
r 3f+3 xix ( 'l-;) xi^i— systems, 
in which x, y, z are not simultaneously divisible by r , render xz — y 2 divisible by 
* It will be seen that 4ij in the Table (i), is in every case the number of the linear forms 8&+ 1, 3, 5, 7, in 
which the numbers M are contained. 
