A formula is given for the number of genera of even lattices with rank
n
n
, signature
s
s
, and discriminant
(
−
1
)
(
n
−
s
)
/
2
d
2
D
{( - 1)^{(n - s)/2}}{d^2}D
, when
d
D
dD
is odd and square-free. In the indefinite case, an orthogonal splitting of these lattices into simple components is also determined.