604 
NUMBERS. 
[795 
being positive or negative even or odd primes, equal or unequal, and similarly 
q, q', q",... being positive or negative even or odd primes, equal or unequal, we have 
P =ppp”... and Q = qq'q". 
.., then the 
symbol 
(1) 
I will 
denote +1, — 1, or 0, according 
to the definition 
\(±\ 
(<£\ 
l/T\l 
(Î\ 
Uv \p) \p’, 
)\p")- 
‘ \pj 
[p'J 
\p’j-’ 
the symbols on the right-hand being Legendre’s symbols. If P and Q are not prime 
to each other, then for some pair of factors p and q we have p = ±q, and the corre 
sponding Legendrian symbol is = 0, whence in this case = 
It is important to remark that 
= +1 is not a sufficient condition in order 
that Q may be a residue of P; if P— 2 a ppp"..., p, p', p", ... being positive odd 
primes, then, in order that Q may be a residue of P, it must be a residue of each 
of the prime factors p, p', p",..., that is, we must have 
as many equations as there are unequal factors p, p', p",... of the modulus P. 
Ordinary Theory, Second Paid,—Theory of Forms. 
26. Binary quadratic (or quadric) forms—transformation and equivalence. We 
consider a form 
ax- + 2 bxy + cif, = (a, h, c) (x, y) 2 , 
or when, as usual, only the coefficients are attended to, = (a, h, c). The coefficients 
(a, h, c) and the variables (x, y) are taken to be positive or negative integers, not 
excluding zero. The discriminant ac — b 2 taken negatively, that is, b 2 — ac, is said to be 
the determinant of the form : and we thus distinguish between forms of a positive 
and of a negative determinant. 
Considering new variables, ax + /3y, yx + 8y, where a, ¡3, y, 8, are positive or negative 
integers, not excluding zero, we have identically 
(a, b, c)(ax + /3y, yx + 8y) 2 = (a, b\ c) (x, y) 2 , 
where 
a' = (a, b, c) (a, y) 2 , = aa 2 + 2bay + cy 2 , 
b' = (a, b, c) (a, y) (¡3, 8), = aa/3 + b(a8 + /3y) + cy8, 
c = (a, b, c) (/3, <5) 2 , = it/3 2 + 2b/38 + c8 2 ; 
and thence 
b' 2 — a'c = (a8 — /3y) 2 (b 2 — ac). 
The form (a', b 1 , c') is in this case said to be contained in the form (a, b, c) ; 
and a condition for this is obviously that the determinant D' of the contained form