264 
PEOFESSOE H. J. S. SMITH ON THE OEDEES 
the four numbers 
(« + l)(a + 2/3 + 1) («+l)(« + 2y + l) 
8 ’ 8 ’ 
(« + 2/3 + 2y + 1) (a + 2/S + 1 ) (a + 2/3 + 2-y + 1) (a + 2y + 1) 
8 ’ 8 
are all congruous to one another for the modulus 2. 
Case (iii) 0 = 1, mod 2, A=2, mod 4. In this case the simultaneous character of 
the forms f and F may be demonstrated as in case (ii), or may be inferred by recipro- 
cation from the result in that case. 
Case (iv) Q.^A=2, mod 4. Let 
A'® = a# 2 +2/3y 2 +4ys 2 , mod 8, 
0'<1> = 4aX 2 +2/3Y 2 4-yZ 2 , mod g, 
a/3y = I, mod 4. 
Here again there are eight cases, 
A'm^a ; O'M^y, or y+2/3 , mod 8, 
A'm= a +2/3 ; 0'M = y, or y+2/3 +4, mod 8, 
A'm = u-l-4 ; Q'M = y+4, or y+2/3 + 4, mod 8, 
A'm = a + 2/3+4; 0'M = y+4, or y + 2/3 , mod 8 ; 
and in all eight the value of the unit ( — 1) 8 8 "F, and therefore of the unit 
/2_ 1+F 2_ ! _ ' ... 
( —1) 8 8 +, is the same, because by virtue of the congruence a+/3+y+l == 0, mod 4, 
the two numbers 
(« + y)(« + y + 2) (« +y + 2/3)(at + y + 2/3 + 2) 
8 ’ 8 
are congruous to one another for the modulus 2. 
Art. 7. The following observations will serve to show more clearly the import of 
the simultaneous character in each of the four cases. 
Case (i) Let "F = — 1 ; then, if m and M are any two uneven numbers simultaneously 
represented by f and F, m = A, mod 4, and M = Q, mod 4. Also f cannot represent 
numbers congruous to 7 A, mod 8, nor F numbers congruous to 70, mod 8; for the 
congruences 
(/3 + l)( 7 +l) _ (y + l)(« + l) ^( « + l) ( /3 + l) ^ 1 
4 4 4 ’ ’ 
imply that a = /3 = y = 1, mod 4 ; i. e. that <p, or, which is the same thing, f can only 
represent uneven numbers congruous to A, 3A, 5A. And similarly of uneven numbers 
F can only represent those which are congruous to O, 30, 50. Numbers congruous to 
3 A are represented by f, and numbers congruous to 30 are represented by F ; but these 
representations are not simultaneous with the representation of any uneven number by 
F in the first case, and by fin the second. 
