Given a family of divisors
{
D
s
}
\{ {D_s}\}
in a family of smooth varieties
{
Y
s
}
\{ {Y_s}\}
and a sequence of integers
m
1
,
…
,
m
t
{m_1}, \ldots ,{m_t}
, we study the scheme parametrizing the points
(
s
,
y
1
,
…
,
y
t
)
(s,{y_1}, \ldots ,{y_t})
such that
y
i
{y_i}
is a (possibly infinitely near)
m
i
{m_i}
-fold point of
D
s
{D_s}
. We obtain a general formula which yields, as special cases, the formula of de Jonquières and other classical results of Enumerative Geometry. We also study the questions of finiteness and the multiplicities of the solutions.